GIÁ TRỊ LỚN NHẤT NHỎ NHẤT

Cty TNHH MTV D W H

ChuySn dg BDHSG ToAn gii trj Idn nhft va g\& trj nh6 nhSX - Phan Huy Khai

Cpng tiifng v6' ba bS't d i n g thtfc tren va c6

Tit (7) (8) va ket hdp vdi x + y + z = 1, ta c6
2

2

2

2

6 + 3 > 3[(xy)^ + (yz)3 + (zx)3

2

2

(xy)3 +(yz)3 +(zx)3

<3

(x + y + z ) + 2 ( x y + y z + z x ) > 3 | x N / ? + y > / ? + z ^ J .

(9)

i Dau b^ng trong (10) xay ra <=> dong thcfi c6 dau b^ng trong (6) (9)

xy + yz + z x < 3 .

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VI

=> x ^

^

2'.

v^s.;

\i

x + 2y^

-= X -

2xy'*

'

. •

= 3y^ ^

(2)

.

up
ro

3y^^x

ok
bo

(4)
(5)

ce

.

•

'

•

,

•

>

,

Z

1

+

+r

x + '

.

(6)

*'
xy + y

isfl tti -

T i r ( l ) (2) suy ra —^
>x + l - - ^

'J—^x + l-——-.
y- + l
?y .
2
Da'u bang trong (3) xay r a o y = 1".

.(purr.

T.v/ix^T.

(x + l)y"

,

(3)
j,

- — ^ > y + 1 - Zillti: ^

(4)

^^'.>,,l_iilii.

(5)

'

2

Da'u b^ng trong (4) (5) ti/dng i?ng xay ra o z = 1; x = 1.

r

Cong ttrng .c-(3) (4) (5) va c6 P > 3 + ^ ^ V ^ ^ - ( ^ y ^ ^ ^ ) .

x + xz + xz > 3x \/z^

O X = y = z = 1.

y + yx + yx > 3 y ^

Do 9 = (x = y + z)' > 3(xy + yz + zx)
=> xy + yz + zx < 3

}S'

1

(6)

Da'u bang trong (6) xay ra <=> dong Ihdi c6 da'u b^ng trong (3) (4) (5)

Lai ap dung bat ddng thtfc Cosi, ta c6

^^'•-.'y.^

+

(2)

x^ + 1

<=> X = y = z = 1.

y

+^
+
y^ +1 z^ + 1 x^

Thco ba'tdangthiJcCosi, taco y ^ + 1 >2y.

+

hayP>3-^(z^ +x ^ + y ^ ) .
3\
Dau bhng trong (6) xay ra o dong thcJi c6 dau b^ng trong (2) (3) (4)

1

(1)

w.

ww

+

+

HUdng ddn gidi
, x+1
, (x + l)y^
Taco:—
=x+l - ' , ^ .

Lap lua n tiTdng tU", ta c6

Dau bang trong (4) (5) tiTdng iVng xiiy ra <=> y = /;\ = x^
Cpng tirng ve (3) (4) (5) va co P > (x + y + z ) - ^Uy[7

X

~-

-

Da'u bang trong (2) xay ra <=> y = 1.

fa

•>z--x\'z

.c

' Dau bang trong (3) xay ra <=> x = y\
2
2
I —
TiTdng tir, CO

— — r - > y - -z^y^ ,
y + 2z3
z + 2x-

(3)

/g

X + 2y''

(•
' •:

om

T i r ( l ) ( 2 ) CO

2xy
2 3rT
-4n= = x - - y V x .

,

Tim gia tn nho nhat cua bieu thifc P =

Dau bkng trong (2) xay ra o x = y\
>x

rr^

""^-'.V'

Thco bat dang thiJc Cosi, thi x + 2y' = x + y ' + y ' > 3 ^ / ^

.

Bai 6. Cho x, y, z la ba so' du'dng va x + y + z = 3.

(1)

x + 2y-

(9)
(lO)

Ta

x'

, _

,j

Vay minP = 1 <!> X = y = z = 1.

s/

Ta CO

: .

o d6ng thdi c6 da'u bang trong (6), (9) o x = y = z = 1.

HUcIng ddn gidi
/

< 3.

xay ra

I

z + 2x^

y + lz^

+

De thay da'u bhng trong (9) xay ra o x = y = z = 1, nen da'u bang trong (10)

Bai 5. Cho x, y. z la ba so Ihifc dufdng va thoa man dieu kien x + y + z = 3.
x + 2y^

+

Bay gic( tiT (6) va (9) di den P > 1.

Nhdn xet: Trong bai tap trcn ta da suf dung k l thuat Cosi ngiTdc!

Tim gia trj nho nha't cua bicu thuTc P =

(g)

TiJf (7) (8) va do X + y + z = 3, suy ra 3 + 2.3 > 3 |x\/z^ + yyfyi^ + z^/y^

?

<=> X = y = z = 1.

o x = y = z = 1.

(7)

Lai do 9 = (x + y + z)^ > 3(xy + yz + zx)

(10)

Tir (6) (9) di den P > 3 - - . 3 => P > 1.

T6mlaitadidenminP=l

Khang Vi$t

,

z

+

zy

+

zy

>

' ' '

3z^.

117


ChuySn 6i BDHSG Toan gii tr| Ifln nhgt

Cty TNHH MTV DWH Khang Vigt

g\i trj nh6 nhS't - Phan Huy KhSi

Da'u bkng trong (7) xay ra o X = y = z = 1.
Do X + y + z = 3, nen tir (6) (7) di den P > 3.
Dafu bkng trong (8) xay ra o dong thdi c6 dau bang trong (6) (7)
<=>x = y = z = l . Vay minP = 3<:i>x = y = z = 1.
Bai 7. Cho x, y, z la ba so thtfc di^dng va x + y + z = 3.
Tim gid tri nho nhat cua bieu thiirc P = -J-— + -^r^— +
x^+1 y ' + l z/ + l
HUdng ddn gidi
(1)
Ta c6: 1 = 1 - x ^ + I

x^+l
Theo ba't dang thuTc Cosi, t h i x^ + 1 > 2x.
(2)
Dau bang trong (2) xay ra <:> x = 1.

(8)

TlM GIA TR! L 6 N NHAT VA NHO NHAT H A M SO

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Phrfcfng phap suT dung bat ding thiJc Bunhiacopski cung la mot trong nhifng
phi/rfng phap cd ban dc tim gia tri Idn nha't va nho nha't ham so' (cung nhuf de
chtfng minh bat dang thuTc noi chung).
;
Giong nhir khi suf dung ba't ding thuTc Cosi, de c6 t h e ap dung mot cdch h i e u
qua phiTdng phap nay, trong moi bai toan cu the can lifa chon mot each thich
hdp hai bo so' roi ap dung bat dang thtfc Bunhiacopski cho hai bo so nay.
Chu y r^ng hai bp so can liTa chon khong doi hoi tinh khong am cua cac so
hang.
„„

Bai 1. (Bat dang thuTc Svcic xd)
' ' *
Cho a i , a2, a^; b|, b2, bi trong do bi, b2, bj la ba so'diTdng. ChuTng minh
a? , , a.^ ^ ( a . + a ^ + a , ) ^
'
'

Ta
up

s/

(4)

y i+1- . l - i .

ro

(5)

ww

w.

fa

ce

bo

ok

.c

om

/g

2
z^ + l
Cong tirng ve (3) (4) (5) va c6 P > 3 - x + y2 + z = -23 (do X + y + z = 3).
Tif do suy ra minP = 3 o x = y = z = I.
Nhan xet: Neu ap dung ngay tiT dau bat ding thufc Cosi x^ + 1 > 2x; y^ + 1 > 2y;
z^ + 1 > 2z, ta co: P < - 1 1 1
(6)
—+ —+ l,x y z
Ap dung baft dang thufc Cosi ccf ban, ta co (x + y + z) —+ —+ - > 9
Tur (6) (7) ta CO P < - ( \— +n— + 2

(7) (do X + y + z = 3)

>1

BUNHIACOPSKI

(3)

Tir(l)(2)suyra - - ^ > 1 - ^ = 1 - J .
x^+1
2x

2
Dau bang trong (3) xay ra <=> x = 1.
Hoan toan tufdng ivt, ta co —

=>- +- +- > 3
x y z

§2. PHLfdNG PHAP SCT DUNG BAT D A N G THLfC

(8)

U yz 2
R6 rang i\i (8) khong suy ra ket luan gi ve moi lien he giffa P va ^ . Phep siJf
dung phiTdng phap Cosi ngifcJc la co hieu qua ro r^t trong bai toan 7, cung
nhiT trong cdc bai todn ttf 1 - 6.

^1

1

1

^2

>

.

,

>

b| + b2 + bj

^.1

[• .,.„. ,.t

Hiidng ddn gidi
- a^
7 ^ thuTc
va Bunhiacop.ski .v d i hai day
Ap diing32ba't+, dang

Vbi" 7^7
(„2

„2

..I

„2^

hj b . (b, + b 2 + b - , ) > ( a ,

Ta co: b|

t...

+32+33)

2

r> u L L ^ . , a? a^ a? ( a . + a j + a , ) ^
Do bi + b2 + hi > 0, nen co: -1- + _i. + _2. > }LJ £ -IL. ^ suy ra dpcm
b| b2 bj b| + b2 + bj
«!

a2

Vb,

7b2

n-'
- ra o \ P i =
Dau bang xay

=

AJ

<z> a,
— = a,
— = a, . 9
b,

b2

b3

tfx

^h^n xet: DSc biet khi la'y bi = aid > 0 (i = 1, 2, 3) ta co ket qua sau:
Cho 6 so a,, 32, 3 3 va Ci, C2, C 3 trong do 3iCi > 0 Vi = 1, 2, 3
vu-

-

3?

-AI

a?

Khi do ta co: —L- + — ^ +
a|C|

a2C2

(31+32+33)^

>
33C3

V

—
3 | C | + 32C2

— .

x

+ a3C3

2. Cho
ba cua
so' thifc
Tim
gia X,tri y,Idnz lanha't
bieudiTdng
thtfc Pthoa
= ^^—-^
xman
+ 1 +dieu
y + 1kien:
+z-+—xyz
1- . = 8. ;

1 IQ


Chuyen 66 BDHSG Toan gii trj lOn nhift va Q\A trj nh6 nhft - Phan Huy Kh5i

Cty TNHH MTV DWH Khang Vi^t
Xa c 6 :

V i e t l a i P diTdi dang sau: P = 3 - 3

r

1
x +l

+

1

"

y+1

L u c nay:

1

-+

-+

1

=^

1

+

Y

vi't^'^^'-r

.

1

-

1

-^97

Z

x^+y

l'f*<ii>>-...- . ' .

1

'

X

Y

P S

;

Z

z^

x^

2 Y Z + Z^

y+1

^

:j

1 ^

(X + Y

+ Z)^

^ (X + Y +

^J

Zf

, (3)

up

T i f do theo (1) suy ra: P < 0.

*

.c

om

B a i 3. Cho ba so' ihuTc diTcJng x, y, z va x + y + z = 1.

/g

V a y max P = 0 o x = y = z = 2.

ro

D a u b i n g t r o n g ( 3 ) x a y r a < = > X = Y = Z o x = y = z = 2.

P>-^—
j +
.
x'' + y ' ' + Z ' '
xy + yz + zx

9

rr

(x + y + z)"^

1

+

\

.

xy + yz + zx

(1 + 1 + 1)2

xy + yz + zx

7

xy + yz + zx

=9

x + y + z + 2 x y + 2yz + 2zx
7

*

+

.

-

i ^ l ,u> A

xy + yz + zx

(3)

'
\

(4)
,
,

•••!.

• •

x=y=z= - .

Bai 4. Cho x, y , z la cac so'thU'c di/dng thoa man dieu k i e n : x y z = 1.
T i m gia t r i nho nhat cua bieu thtfc P = - r - ^ ^
+
+-r-^
.
x ' ( y + z) y ' ( z + l ) z-\ + y )

1

I jXf:

1
(1)
i

A p dung bat d^ng thuTc Sv^c-xd tuT (1) c6:
1 1^'

'I

•

—+ — +

P>

(2)

X

y

-

(xy + yz + zx)2

z

2(xy + yz + zx)

2.2

(2)

2

2(xy + y z + zx)x^y^z

Do xyz = 1, nen tir (2) ta co: P > ^y + y^- + ^^ .
DSfu b i n g trong (3) xay ra <=> x = y = z = 1.

D a u b i n g trong (2) xay r a o x = y = z = j

+ -

Nhif vay m i n P = 30. Gia t r i nho nha't dat di/dc k h i x = y = z = ^ .

ok
:''

D a u b i n g trong (1) xay r a o x = y = z = - .
Tiir(l)fac6:

,

x y + yz + zx

1
:2

ce

fa

.,

xy + yz + zx

v2
12
Tac6:P = —
+
+—^
.
x ( y + z) y(z + l ) z(x + y )

w.

fi

ww

x+y+z

+

Hudngddngidi

bo

xyz

Ta c6: xy + yz + zx = xyz — + — + - > xyz
= 9xyz
X
y z ]
x+y+z

xyz

+z

1

TO (3), (4) suy ra P > 30 v i dau b i n g xay ra o

Z + 1 " 2 X Y + Y 2 + 2 Y Z + Z ^ + 2 Z X + X ^ ~ ( X + Y + Z)^

T i m gia t r i nho nha't cua bieu thiJc P = - r — \ ^
x^+y^+z^

,

Ta

x+l

1

xy + yz + zx

1
Dau bang trong (4) xay ra o X = y = z = - .

s/

Bunhiacopski) ta c6:
^

T+

=:>xy+ yz + zx < - = >
>21.
3
xy + yz + zx

2ZX + X ^

A p dung bat dang thiJc Svac-xd (Bai 1 - dang dac biet cua baft dang thiJc

1

^
+ y +
z

1

L a i c6: 1 = ( X + y + zf = x^ + y^ + z^ + 2(xy + yz + zx)

" 2X + Y ^ 2 Y + Z ^ 2 Z + X
2XY + Y^

=x

1

DSu b i n g trong (3) xay ra o x = y = z.

X

^

xy + yz + zx

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1

- +

1

.jji^vH

X p dung baft ddng thuTc Svdc-xO ta c6:

Y>0,Z>0(vlrorang
.-pi/iq

,

9

P + y1^ + z

xyz = 8)
.

1

z + l.

9Y
9Y
77
Datx= — ; y= — ; z -—.Khid6tac6:X>0,
Y
Z
X

,

'

. v*-

L a i theo bat d^ng thtfc Co Si ta c6:
xy + yz + zx >3^{\yzf

•

i>
;

=3.
• •

(3)

(4) (do xyz = 1 )
•

.

• "

•

'

1 T

1


Cty TNHH MTV DWH Khang Vigt

ChuySn dg BDHSG To^n gii tri lOn nha't va gia tr| nh6 nhS't - Phan Huy Kh^i

Dau bang trong (4) xay ra <=> X = y = z = 1.

Ta c6:
|,4

y.^^r^—^^

;

Da'u bang trong (5) xay ra <=> dong thdi c6 dau bang trong (3)(4)
o x = y = z = 1.

.

.:•

V a y m i n P = ^<=>x = y = z = 1.

-

'

v

^• .

Bai 5. Cho x, y, z la ba so thiTc diTcJng thoa man dieu kien: xyz = 1 .
X
y
z
Tim gia trj nho nhat cija bieu ihuTc P =
- +
+
•
2 , 2 , 2'
M
1+ - 1+1+ iX\

z

X

*

" i >

(3)

X

w.

«

i

Bai 6. Cho x, y, z la cac so thifc difdng ihoa man dieu kien: xy + yz + zx = 4.

Ap dung bat dang thuTc Bunhiascopki cho hai day s6':
•

^: x ^ y ^ z ^
1,1,1.

•

y

Z

xy + yz + zx = 4
X , y, z cung dau

2

= xy

c

^ <

3

o
x=y=z=

-2N/3

Tit do suy ra min P =

2v^ . „

2V3

B i i 7. Cho X , y, z la cac so thuTc diTdng thoa man dieu kien: x + y + z = 3.
x-^
y+z
Hudngddngidi

= y = z = 1.

Huditg dan giai

2

y = xz

16 • '•'•^'^
a-.

(4)

i t

Tim gia tri nho nha't ciia bieu thiJc P = x"* + y"* + z*. ^

X

Tim gia tri nho nha't cua bieu thtfc P =

ww

o X = Y = Z<=>x = y = z = l

x^ = yz

2V3

Ta
s/
up
ro

/g

om

bo
ce

' •

Dau bang trong (4) xay ra <=> dong thdi CO dau bkng trong (2)(3)
Vay min P = 2 <=>

y

Gia tri nlio nha't dat difdc khi x = y = z = — j - hoSc l a x = y = z = - - ^

fa

X = Y = Z.

Tiif (2), (3) co: P > 1.

X,

(2)

ok

~.
2(XY + Y Z + ZX)2

D e t h a y r k n g ( X + Y + Z ) ' > 3 ( X Y + YZ + ZX).
'

z,

Dafu b^ng trong (3) xay ra khi va chi khi

.c

(X + Y + Z)^

Dau bkng trong (2) xay ra o X = Y = Z.
Dau bang trong (3) xay ra

y, z

lflf(l),(z;suyrax +y +z

XZ+2YZ

' '
Ap dung bat d^ng ihufc Svac-xd, ta c6: P >

X,

z

'

(chu y la xyz = 1, ncn c6 the ddi bien nhuTtren)
Z
X
Y
X
Y
Z
Y
1

Z +
^
•
+
, 2Z , 2X
Y + 2Z Z + 2X + -X + 2Y
1+—
1+ —
X
i
Y
Z
Y^
X2
±
(1)
..orfJ.-

+ ZY+2XY

y^ + z ^ f ^ 3(x^ + y'* + z'*) > (x^ + y2 + z^ f .

H-

x y z

Xet phep ddi bien x = —; y - — va z = —, vdi X, Y, Z > 0.
Y
Z
X

XY+2XZ

) ( l ^ +1^ +1^ ) > (x^

Taco: ( x ^ + y ^ + z ^ ) ( z ^ + x 2 + y 2 ) > ( x y + yz + zx)^ = > ( x ^ + y ^ + z ^ ) > 1 6 .

!

HUdngdangiai

=

+

Dau bling trong (1) xay ra o x^ = y^ = z l
Lai ap dung baft d i n g thiJc Bunhiascopki cho hai day so:

.

J

^ '

+

(5)

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Tir (3), (4) suy ra P > I.

,

+

——
z+x

+•

x+y
-Jwrm^

z^
x"*
v'*
z"*
Tac6: p = J— + ^
+^
=—

+^
+—
. (1)
y + z z + x x + y xy + xz yz + xy xz + yz
(x^+y^+z^f
Ap dung bat d i n g thuTc Svac-xd, ta c6: P > ^
'-- . (2)
2(xy + yz + zx)
Da'u bkng trong (2) xay ra<=>x = y = z = l (dox + y + z = 3)
D l tha'y r^ng: x^ + y^ + z^ > xy + yz + zx.

(3)

Da'u bkng trong (3) o X = y = z = 1.
123


Chuy6n dj BDHSG Toan g\& tr| lOn nha't

T i r ( 2 ) , ( 3 ) t a c 6 : P>^5^

^

Uiin nhien ta c6:

+

+

Cty TNHH MTV DWH Khang Vi^t

g i i trj nh6 nhgt - Phan Huy Kh5i

- .
>

V a y m i n P = 2 <=> x = y = z = ^ .

(4)
^^

.

Cac ban hay so sanh cdch giai nay v d i cdch dung phiTdng phap them bdt

(5)

hang t ^ va bat dang thiirc Cosi.

Dafu b^ng trong (5) <=> X = y = z = 1.
(6)

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T i m gid trj nho nhat cua bieu thtfc P =

Dau bkng trong (6) xay ra o dong thdi c6 dau b^ng trong (4), (5)
X

d p dung bat d i n g thiJc

X

(*

X,

y, z > 0

VI X

bo

o

• 1 ,<

^
3

\2

=

2

J

2
2
^\ i .

6

(4)

6

X =

y =z =

3

ww

fiai 9. Cho x > 0, y > 0, z > 0 va x + y + z = 3.
x^
x+y

z^

j

+— —+
y+z z+x

T i m gia tri \dn nhat cua bieu thi?c P =

X^

+zf

A p dung ba't d^ng thuTc Svdc-xd, ta c6: P > ^ ^ ^ / ^ ^ ^

x + +z
= \ y+2 ^

1

V i e t l a i P duTdi dang sau:
4

p=

I
x-''+2xy

4

4

y^+2yV

y2

^2

r- + — - — +
x + 2y^
y + 2z^
z + 2x^

HUdng dan giai

L^i giai nhu sau:
(\+

(0

(2)

^
3

Vay m m P = - o x = y = z = — .
6
^
3

+ y + z = 1.

T i m gia t r i nho nhat cua bieu thiJc P =

•

DSu bSng trong (4) x a y ra o ddng th6i c6 dau b i n g trong (2), (3)

Tilf d6 suy ra m i n P = ^ o x = y = z = 1. Ta thu l a i k e t qua tren.

Cho

.

I

6(x2+y2+z2)

fa

2'

tlci-ivi';::!;'•

M d c khac ta c6 x y + yz + zx < x^ + y^ + 7?.

ce

_3

2. Ta c6 b a i toan tuTdng tif sau:

.

x^+y^+z^)
Tit(2), (3) ta c6: P > - L —J-

w.

6

1

Ta
s/
up

D o x^ + y^ + z^ > x y + yz + zx, nen tif (*) suy ra:

.c

> x^ + y^ + z^.

ok

om

/g

>z2.

Cong tuTng ve 3 bat dang thtfc tren ta c6: P + — — ^

2

^
z + 2x + 3y '

+ y + z + 5(xy + yz + zx)

Dau b i n g trong ( 3 ) o x = y = z =

ro

4

x^+y^+z^ ^ ( x + y + zr

4 '

DSu b^ng trong (2) xay r a o x = y = z =

, y(z + x )

x +y

1—

p>

Cosi de giai b ^ i toan tren nhtf sau:
x(y + z)

x^
Taco: - ^ + " ' ^ " "
,
4
y+z

z(x + y )

^
•
^
+ 2y + 3z y + 2z + 3x

X

Ap dung baft d^ng thufc Svdc-xd, ta c6:

v.

1. Ta CO the suT dung phiTdng phap them bdt hang tuT

Z + X

s HO ' .

T a c 6 : P = ^—-^^^
^ - +- 5 — T
^ +
T
T"X + 2 x y + 3xz y + 2 y z + 3xy

z + 2 x z + 3yz

^^ajV

¥?f -

'

m .(i;xrri

4

4

Vay m i n P = - o x = y = z = 1.

' '

Hudng ddn giai

= y = z = 1.

Nh^nxet:

' " " • ' ''

p ^ l 8. Cho X > 0, y > 0, z > 0 va x^ + y^ + z^ = 1 .

Tilfx + y + z = 3 . ( 4 ) , ( 5 ) s u y r a : P > | .

o

^

z^+2xV

(3)


BDHSG Toan gii trj Idn nhSt va gia tr| nhd nhat - Phan Huy KhSi

Cty TNHH MTV DWH Khang Vi$t

Theo baft dang thtfc Svac-xct, ta c6:
,

Dau bang trong (1) xay ra

H'- i'sifn V >

••• \ -f.

Da'u bang trong (1) xay ra o x = y = z = 1 (do x + y + z = 3).
x^+y'*+z^+2(xy+yV+zV)
Tac6VP(l)=
,
,
,
) , 2
2 2

2 ,

b

c

ax

bx

cz

,,

1
x

1
y

1

1

z

3

1

Ti/dng t y co:

; .>p .

.

do —+ —+ - = 1
X
y z

I

b c a'
—+ —+ X
y z

I

c a b
—+ —+ —
X
y z

bx + cy + az
1

(2)

A p dung baft dang thtfc Cosi ta c6: x'* + x"* + x > 3x^

a

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Chuy6n

cx + ay + bz

<=>x = y = z = 3.

(3)

Dafu b^ng trong (2), (3) deu xay ra o x = y = z = 3.

<,,

• h

C6ng tCfng ve (1), (2), (3) va c6:

z* + z*

z* + zz>>33zz' \
. ^

—

ax + by + cz

Cong tijfng ve 3 bat dang thuTc tren ta c6
2 ( x ^ + y ' * + z ^ ) + (x + y + z ) > 3 ( x - U y U z ^ ) .

*

y H l + l>3y
- \
infid v..

.1

..,

up

ok

ww

B a i 10. Gia suf x, y , z la cac so thiTc di/cfng sao cho - + - + - = 1 .
X
y z
'

Cho a, b , c la ba h^ng so difdng cho triTdc. T i m gia t r i Idn nhat cua bieu thiJc
P=

I

1
+
ax + by + cz
bx + cy + az

1

c^

(a + b + c)^

ax

bx

cz

ax + by + cz

^ 1 ^a b c
—+ —+ ax + by + cz
^x
y
z,
1

1 P
fl 1 P
+ - + - + c —+ —+ U
y z;
Ix
y 'L)

1
ax + by + cz

1
bx + cy + az

1

1

cx + ay + bz

a+b+c

DSu bkng trong (4) xay ra o dong thdi c6 dau b i n g trong (1), (2), (3)
O x = y = z = 3.

max

?
o x = y = z = 3.
a + b+ c

1
2 x + 3 y + 6z

1
3x + 6y + 2z

1
= 1
6x + 2y + 3z
{St'

n^unhir - + - + - = 1 v i x > 0 , y > 0 , z > 0 .
X
y z

rim

11. Cho X, y, z la cdc so diTdng.
^
y + 2z

z + 2x

-+•

x + 2y

Hudng ddn giai

D i l t S = a + b + c > 0. A p dung bat d^ng thi?c Svdc-xd, ta c6:
b^

'

+ bf -l

T i m gia trj Idn nhS't ciia bieu thuTc P =

cx + ay + bz

HUdng ddn gidi
a^

cx + ay + bz

Ap dung: N e u a = 2, b = 3, c = 6 ta c6 ket qua sau:

Olm^h

w.

V a y min P = 1 <=> X = y = z = 1.

'

(6)

fa

Dafu bkng trong (7) xay ra de tha'y o x = y = z = l

bo

Tiir(l),(6)suyraP>l.

/

i ' "

V$ymaxP=

ce

.

1

Ta
s/
(4)

Tir (3), (4) suy ra: x" + y^ + z" > x U y ' + z\ s,\)
T u r ( 2 ) , ( 5 ) c 6 : V P ( 1 ) > 1.

bx + cy + az

+

.c

'

,.

om

Do X + y + z = 3, nen c6: x^ + y^ + z' + 2(x + y + z) > 3(x + y + z)

ro

3|di'a/'

/g

Tijf do suy ra: x ' + y^ + z^ + 6 > 3(x + y + z). '"'^ +

De thay dau b^ng trong (5) o x = y = z = 1.

1

Tif gia thiet - + - + - = \a S = a + b + c suy ra:
X
y z

+ 1 + 1 >3z

*

+

(\ r
a —+—+U
y

(3)

Theo bat d^ng thuTc Cosi ta lai c6: x^ + 1 + 1 > 3x

=>x^ + y^ + z ^ > x + y + z.

1

(1)

' i ^ t l a i P dirdi dang sau: P = — - — + — ^ — — + — - — .
xy + 2xz
yz + 2xy
xz + 2yz
- (1) va theo ba't d i n g thtfc Svac-xd, ta c6:

(1)


Chuy6n

P>

BDHSG Join gii trj Idn nha't va gia trj nh6 nh3"t - Phan Huy Khii

(x + y + z ) !
hayP>
x y + 2xz + yz + 2 x y + xz + 2yz
^ ., ^

Cty TNHH MTV DWH Khang Vigt

(^^y^^-)'
.
3(xy + yz + zx)

/. = x

(2)
i.

Ro rang (x + y + z)^ > 3(xy + yz + zx), nen tijf (2) suy ra: P > 1.

Dau b i n g trong (3) xay ra <=>

Nhdit

X

Uj'iih

•}!;:>

'ixil,

^2y^(l-y^)^(l-y>-^VV^4(l-2y^).l.

Tacd:

= y = z > 0.

xet:

(4)

L a i ap dung ba't dang thiJc Bunhiacopski v d i hai day:

d i n g thufc Bunhiacopski.

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1. Cach giai tren la diTa vao bat d^ng thtfc Svac-xd (mpt dang dac bi^t cua bat
'

2. X e t each g i a i sau day diTa vao ba't dang thiJc Cosi
Dat X = y + 2z; Y = z + 2x; Z =

.

. .

|

X > 0, Y > 0, Z > 0
4Z + X - 2 Y

4X + Y - 2 Z

; y=

Y

; z=
9

4Z+X-2Y
9Y
9

9Y

9Y

^/3.

9

4X + Y - 2 Z
9Z
9

Y^
1
Z^ 1 f z
fx
—+
+
—
—+
—
+—
9
9
9
Iz
Ix
1

9Z

j

9Z

X^
Zj

9

Z

1 (Y

+ 3- —+ —+

IZ

Y

x^
zj

2

(*)

3
hayP>l.

(**)

.c

om

Theo bat dang thurc Cosi, t i i f ( * ) suy ra: P > ^ + ^ + -^ + l - |

ok

De thay dau bang trong (**) x a y r a < : : > X = Y = Z > O o x = y = z > 0 .
= y = z.

bo

Vay min P = l o x

ce

D o do ro r^ng phiTdng phap giai sijT dung bat dang thuTc Svac-xcf la gon gang

w.

B a i 12. Cho x, y, z thoa man d i c u k i e n : x^ + y^ + z^ = 1.
T i m gia t r i Idn nha't cua bieu thiJc P = xy + yz + 2zx.
Hudng ddn giai

•xy + y z < |x^
^ +z^^

De thay: 2zx < x^ + z^ = 1 -

(2)

P<^2y2(l-y^)+(l-y^).

(5) - J M g n ; ; )

(6)

Da'u bang xay ra trong (6) o
X

2V

V2V4y2-4y^:.2(l-2y2).
+ y ^ +z^ = 1 • ; •

X

'

d6ng thdi c6 da'u b l n g xay ra trong (3), (5)

=z

2

(7)

^/3

2^/3

(8)

x, y, z th6a m a n he (7), (8).

<. , r ,

> 0, y > 0, z > 0 va x + y + z = 3.

(*)
(1)

Tiir(l),(2)suyra:

1

l + 2y-'

Do x^ + y ^ + z^ = 1, nen tiir (*) suy ra: xy + yz < ^ 2 y ^ ( l - y ^ ) .
y\

• .

4j

T i m gia t r i nho nha't cua bieu thtfc P =

A p dung bat d i n g thuTc Bunhiacopski, ta c6:
xy + yz < Vx^ -hz^.yjy^ +y^

,

JU

Tir (3), (4), (5) suy ra: P <

^> 13. Cho

ww

r

1

Tiir do ta c6: max P = ^ ^ l l »

fa

hcfn each suT dung bat dang thtfc Cosi.

P
(4y2-4y'*) + (l-2y2)^ Y i—+ —
>

Ta

9
4Y + Z - 2 X
9X

s/

vax=

talaico:

up

4Y + Z - 2 X

9X^9

,jr

ro

CO

2y.

X +

/g

K h i do ta

„

<=> Z = X

(3)

De thay dafu bkng trong (3) xay ra <=> x = y = z.
Vay min P = 1 o

X _ y

(3)

l + 2z^

l + 2x-^"

Hudng ddn giai

,

Theo bat dang thii'c Svac-xd, ta c6: P >
^ ^ ^ y + ^)^
"'3 + 2 ( x ^ + y ^ + z ^ ) C>au bkng trong (1) xay ra o

/n

x = y =z= I

Theo bat d i n g thtfc Cosi, ta cd:

x^ + 1 + 1 > 3x
y^ + 1 + 1 > 3y

>'•••''

^^f>}b "\ •:

'hv

1

f\C\


Chuy6n dg BDHSG Toan g\A tfj lOn nha't

Cty TNHH MTV DWH Khang Vi$i

gia-trj nh6 nhat - Phan Huy KhSi

z ' + 1 + 1 > 3z.
TiJf do va difa vao gia Ihiel x + y + z = 3 suy ra:

+

+ z^ > 3.

P =

(2)

Dau bang trong (2) xay ra<=>x = y = z = l .
(3)

P>

Da'u b^ng trong (3) xay ra <=> dong thcfi c6 dau bang trong (1), (2)
,;„.,.:u

„vi

,.,::;f,

.VM'M-.r,

B a i 14. Cho x, y, z la cac so' thiTc duTdng. T i m gia t n nho nha't cua bieu thtfc

y +yz + z

:r
2 + -^2

T +

2

2

x +xy + y

z +ZX + X

2 •

'

HUdngdangidi
'..i

^

X

x ^ ( y ^ + y z + z^)

y

^

y ^ ( z ^ + z x + x^)

z

z?(x^ + xy + y ^ )

A p dung ba't dang thuTc Svac-xd, ta c6:
2\

up

x^+y^+z^

hayP>^-—

/g

x^+y^+z* + 2 f x y + y V + z V )
, ,
,

^

'
(2)

,

bo

Theo bat d i n g t h i J c C 6 s i , t a c 6 :

ok

2(x^y^ + y^z^ + z^x^ J + ( x y ) ( y z ) + (xy)(zx) + (yz)(zx)

om

.

ro

i
L
x^(y^ + y z + z^] + y^(z^ + z x + x^) + z ^ ( x ^ + x y + y^)

.c

p>

, , _

s/

2

1 2

«, ^

ce

x ' + y^ + z ' > x y + y V + z V

(3)
(4)

w.

z'• + z* + z > 3 z ^

ww

sfx^y^+yV+z^x^]
Tiir (2). (3), (4) suy ra: P >
—
— ( hay P > 1.
,
3(x^y^+yV+z^x''j
D c tha'y da'u b i n g trong (5) xay ra o
'

(5)

x = y = z > 0.

V a y m i n P = 1 <=> X = y = z > 0.

i i a i 15. Cho x, y, z la cdc so thiTc diTdng. T i m gia t r i nho nhat cua bieu thiJc
„

P,=

X

yjx^+Syz

+

,y
z
/
+ /
7y^+8zx
^jz^+8\y
HUdng dSn giai

(2)

-i,;-

= (x + y + z ) ( x - ' + y ^ + z ^ + 2 4 x y z ) .

; . : :•

A p dung bat dang t h u t Cosi, ta co:

v^ r <

(3)
\

(X + y + z ) ' = x ' + y^ + z ' + 3(x + y + z)(xy + yz + zx) - 3xyz

ri

>x^ + y ' + z^+ 2 7 7 x y z . ^ / x y ? - 3 x y z

v,

hay (X + y + z)^ > x^ + y^ + z^ + 24xyz.

.t*-->.

A . ' . , - y g „ s f : r X) < x

(4)

Thay (3), (4) vao (2) va c6: P > ^ ^ i i ^ ^ i ^ = 1 .
(x + y + z r
Dg tha'y dau bling trong (5) xay ra o

(5)

x = y = z = 1.

A^Aa/i A:^^; Ta c6 bai tocin tu'dng tu" sau:

\.T:

«Vt

Cho X > 0, y > 0, z > 0 va x + y + z = 1. T i m gia tri nho nha't ciia bieu thtfc
P =

fa

x^y^ + fz^ + z V > (xy)(yz) + (xy)(zx) + (yz)(zx)

x7x^ + 8 y z + y^jy^ +8zx + z^z^ + 8 x y

(x + y + z) x ( x ' + 8 y z j + y ( y ^ + 8 z x ) + z ( z ^ + 8 x y j

.

V i e t l a i P diTdi dang:

(x + y + z ) -

V ^ . V x ^ x ^ + 8 y z + ^/y.^/y^/y^ +8zx + VZ.N/Z^Z^ + 8 x y J

Ta

2

(1)

z^z^ + 8xy

A p dung ba't dang thuTc Bunhiacopski, ta c6:

Vay min P = l < = > x = y = z = l .

P=

yyjy^ + 8zx

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= y = Z=l.

+ 8yz

Til' (1) va theo bat dang thifc Svac-xd, ta c6:

Tird), (2)suyraP>l.

O X

i l l . h:l

x7x^

x'^+Syz

y^+8zx

z^+8xy

Ta giai nhi/ sau: P = -j-^
+
X +8xyz
y +8xyz
A p dung bat dilng thuTc Svac-xd, ta c6: P >

z2

z +8xyz
^ (x + y + z)
x' + y +•/: + 2 4 x y z

Theo bai tren ta c6: (x + y + z ) ' > x ' + y^ + z^ + 24xyz.

(*)
(**)
(***)

Tir (**)^ (***) suy ra: P > ^^^^^^'^^ hay P >
^
= 1.
(x + y + z)
x+y+z
1
Vay min P = l o x = y = z = - .
^aj 16. Gia sur x, y, z la ba canh cua mot tam giac c6 chu v i bang 12.

V i e t l a i P dtfdi dang sau:
Ill


ChuySn 6i BDHSG Toan glA trj lOn nha't

Cty TNHH MTV DWH Khang Vigt

glA trj nh6 nha't - Phan Huy KhSi

pai 18. Cho x, y, z la ba so'dtfcfng thoa man dieu kien:

, .
y+ z- x
/ +x - y
x+y - z

Tim gia tri nho nhat cua bieii ihiTc P =
+
+
•
3x + y - z 3y + z - X 3z + X - y

x(x- l) + y ( y - l) + z(z- 1) < ^ .

Hiidlng ddn gidi
Dat X = y + z - x; Y = z + X - y; Z =

X

Tim gia tri I6n va nho nhat eua bieu thufc P = x + y + z.

+ y - z.

HUdng ddn gidi

Khi do ta CO X > 0, Y > 0, Z > 0.
;y =

Z + X

2

;z=

Viet lai gia thie't da cho diTdi dang:

X + Y

3 ( x ^ + y ^ + z 2 ) < 3(x + y + z) + 4.

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De thay x =

Y + Z

Tiif do ta c6:
P =

Ap dung ba't ding thuTc Bunhiacopski, ta c6:

Z

(X

Y + 2 Z ^ Z + 2 X ' ^ X + 2Y

X Y + 2XZ

Z Y + 2XY

X Z + 2YZ'

( X + Y + Z)^

De thay P = 1

'

X = Y = Z o x = y = z = 4. Vay min P = 1.

Gid tri nho nhat khi x, y, z la canh cua tarn giac deu co chu vi bang 12

x';

ro

HUtUg d&n giai

om

1

>— 3+x^
2

j =

x = 3.

11

Tif(l)tac6: x + — + ,
2x V
X+ -

x

>

6

1+

>X +

x^

ww

11

3

7

+ - +

2x

2

2x

(1)

+ y + z)^ - 3(x+ y + z) - 4 < 0

- l < x+ y+z < 4 o - l < P < 4 .
o

(3)

x+y+z=—
4
3 o x +y+ z= - .
x =y= z
1

TiTdng tu'P = - l c i > x = y = x = - - .
4
1
Vay max P = 4 o x = y = x = —; min P = - l o x = y = x = —
3

.

3

sCf DgNG BAT D A N G THLfC

D E TIM GIA TR!

L6N NHAT VA

NHO NHAT CUA HAM SO
1. Phrfring phap xua't phap tuT mpt bat dang thuTc da bie't tijf trtfdfc.
PhiTdng phap xuat phap ttr mot bat dang thiJc nao dafy da diing, sau do bicn

3

9

=:- + X + 2
X

doi thanh bat dang thtfe dang P ^ a

(1) (hoae P < a), d day P la bieu thiJc

can tlm gia tri nho nha't (hoac \dn nha't).
Sau khi chi ra phan tur da cho tfng vdi phan tur do thi P dat gia tri a, ta se suy
ra ke't luan min

<,

P = a (hoac max P = a).

Nhir vay dieu co't ye'u khi sur dung phiTctng phap nay la can lira chpn cac bat
15

Dau b^ng trong (2) xay ra o x = 3. Tif do suy ra: t(x) > y .
Dau bang trong (3) xtiy ra o ddng thcli c6 dau bang trong (1), (2) o x = 3.
Vay min f(x) =

•i

§3. CAC PHadNG PHAP THONG DgNG KHAC

.c

Xy

w.

Dau bang trong ( 1 ) xay ra o

V

bo

1+

3.1

>

ce

. 4 . | l + - ^ > 3 + - =>
x^
X

x';

ok

1+ -

V

fa

'
7 W - 7
> 3+ (9 + 7) 1 + V X' J

2

o

/g

Ap dung ba't dang thtjTe Bunhiacopski, ta eo:

(2)

'

(X

Ta

1+ -

V

up

V

s/

+

2x

> x;,

Z^) .

o

Tif do suy ra: P = 4

(tinh X = y = z = 4).
Bai 17. Cho x > 0. Tim gia tri nho nhat cua ham so l(x) = x +

+ y + z)^ < 3(x^ + y^ +

TiJf ( 1 ) , (2) suy ra: (x + y + z)^ < 3(x+ y + z) + 4

^,

Tilf ( 1 ) va thco bat dang thuTc Svac-xd, ta c6: P > 3 ( X Y + Y Z + Z X )

Do

(1)

y

<=> x = 3.

dang thtfc thich hdp vdi de ra de c6 the bien doi ve ba't dang thiJc dang ( 1 ) .
Viec lira chpn nay du'pc tien hanh b^ng each diTa v^o cau true cua bieu thtfc
• P ban dau cung nhtf cac gia thiet cua bai toan.
P^i 1. Cho X, y, z \h. ba so diTtfng va thoa man dieu kien xyz = 1.


Cty TNHH MTV DWH Khang Vigt

Chuyfin dg BDHSG loin g\i trj \6n nhi't vS gia tr| nh6 nhSft - Phan Huy KhSi

T i m gia tri Idn nhat ciia bicu thtfc P = — 5 — ^
+——
xUy'+l

y^+z^+l

+

=> x* + y'* + 7* > xyz(x + y + z).

— ^
.
z-'+x-Ul

Nhan xet dMc ch^ng minh: De tha'y dau bang trong ( 1 ) xay ra <=> x = y = z.
Do xyzt = 1, nen ttr (1) c6:

HUdng ddn gidi
H i e n nhien ta c6 bat dang thtfc sau:

- xy +

> xy.

I

(1)

*

I

Dau bkng trong (1) xay ra o x = y * H«*'
(x + y ) ( x ^ - x y + y ^ ) > x y ( x + y )

=> x^ + y ' > xy(x + y).

' \

1

x ' + y % 1 > xy(x + y + z)

Lap luan tiTcMg tiT ta c6:

"
.

'
<
!
.
x"* + y"* + 1 xy(x + y + z)

i

n

;

fMJ"''

^

(3)

—
3—- < 7 3 7 7 - 7 — 3 ; .
y-^ + z-^ + 1 yz(x + y + z)

(4)

Ta
up
(

6

ro

.

)

/g

^^^^^ =— =1
xyz(x + y + z) x y z

=y=z= l.

ce

fa

"1

P W'

vho ca'u true cua bieu thtfc P.

Bai 3. Cho x, y , z la ba so diTdng va xyz = 1. T i m gia tri Idn nha't cua bieu thiJc
2 2

1

X** + y " + z* > xyz(x + y + z).

^^^^
'

7

7

7

y^z^'+y'+z'

'^^

2

2

7

7'

z^x'^+z'+x'

HU^ng ddn gidi rll r.:X.,X'''-':^>
5

u,ul^k;

x V + x^ + y^ > x V + xV'(x + y)
xV
xV

Tir (1) co:

^^^^^-^^^^

'
(1)

That vay: x U y " + z^ > x V + y^z^ + z^x^
=> 2(x^ + y^ + z^) > (x^y^ + y^z^) + (y^z^ + z^x^) + (z^x^ + x^y^)
> 2xy^z + 2xyz^ + 2x^yz

2

(1)

Dau bling trong (1) xay ra o X = y.

1

l^zSt^+x^/t^+x^+y'^+r

TrU'dc het ta chi?ng minh rang v d i m o i x, y, z ta c6:

7

( x ^ - y ^ ) ( x ^ - y ' * ) > 0 = o x ^ + y ' > x V ( x + y).

^

fj^^„gj^,^gi^i

7

x''y^ + x ' + y '

Do X, y 1^ cac so di/dng nen ta c6:
"

w.

1

- x ^ + y ^ + z ^ l ^ y ^ + z ^ + t^ +
vt.^a..i.,MiiMal.b''-5;rvi6

ww

' '

(6)

- j ' ' ' ' " " " ' " • ^'

B a i 2. Cho x, y , z, I la bon so thiTc dtfdng sao cho xyzt = I .
T i m gia trj Idn nhat cua bieu thiirc

(5)

r . *-v i r ' , (

•

^ ^ " 8 Ihtfc (1) trong bai 1 nhi/ng viec suT dung no cung la le tiT nhien v i diTa

J»uH.». v)¥\m.i ^lAS © M I I G

(x - y)^ > 0 i=> x^ - xy + y^ > xy.

,,,-;Af.tt,.

.1;..

-(4)

I ^

A^/iflH xet: Trong bai nay ta da su" dung bat dang thiJc (1), do la tiif bat d i n g thiJc
h i c n nhien

^

:/ ; i r

..-.Hhihkt.Mhlt.,

Vay max P = l o x = y = z = t = I .

bo

At Way max P = l o x

<=> X = y = z = 1 (do xyz = 1 )

i4 U :i .'I

:5 m

(3)

Nh4n xet: 6 day ta difa v^o b5't d i n g thiJc (1). N o tuy khong ddn gian nhiT baft

ok

*>..?-t.i's/f w f r v

^

,

De iha'y dau bang trong (6) xay r a o x = y = z = t = l .

.c

om

Dau bang trong (6) xay ra o dong thdi c6 dau bang trong (3), (4), (5)
.

<

Dau bang trong (3), (4), (5) tiTdng iJng xay r a o y = z = t ; z = t = x ; t = x = y.

zx(x + y + z)

c6 P <

—

Cong tirng ve (2), (3), (4), (5) va chii ^ xyzt = 1, ta c6: P < (1).

Da'u bkng trong (4), (5) liTdng uTng xay ra k h i y = z; z = z
Cong ti^ng ve'(3), (4), (5)

(2)

y + z +1 +1 yzt(x + y + z +1)

t + x + y + 1 txy(x + y + z + t)

-(m

— .

xyz(x + y + z + t )

Ztx(X + y + Z + t)

1

s/

z'+x^' + l

„ ji-i

—

-

Z4^j4^^4^1

Dau bang trong (3) xay ra o dau bang trong (1) xay ra o x = y.
TiTdng tur la c6:

L

Dau bang trong (2) xay ra o x = y = z

(2)

-,

—-<
< ——

x'* + y"* + z"* + 1

;

D o x y z = l , n e n t i r ( 2 ) c 6 : x ^ + y ' + 1 > x y ( x + y) + xy/

J

Do X, y, z, t > 0, nen ta c6: —

iL
ie
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nT
hi
Da
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oc
01

/

V i x > 0 , y >0,nenlil'(l)c6

+ 1 ^ xyz(x + y + z) + xyzt => x'* + y"* + z"* + 1 > xyz(x + y + z + t)

x"* + y'* +

x^y^+x^+y^

' ''
hay

x^y^+x^y-Xx + y )

•'
x^y^ + x ^ + y ' '
-'y'

x^y^+x^+y^

<

'

• '

„

1

1

l + xy(x + y )

xyz + xy(x + y )

'

xy(x + y + z)

.

/

>^ ^

•

(2)

4Vt
135


ChuySn dS BDHSG Tpan g\& trj Ifln nhSit vS g\i trj nh6 nha't - Phan Huy KhSi

Cty TNHH MTV DVVH Khang Vi^t

D a u bang trong (2) xay ra <=> x = y.

TufOng tU ta c6:

Dox + y + z = 4 = > P < 8 .
1

y^z^

+z'^

yz(x + y + z ) '

Dc thay da'u bang trong (4) xay ra

(3)

^'

<=> trong 3 so c6 mot so bang 0, hai so bKng 2.

(5)

NhiT the max P = 8 o x, y, z thoa man (5).

zV+z7+x^

(4)

zx(x + y + z)

* ' "

-

D c thay dau bang trong (5) xay ra<=>x = y = z= 1 . , y i

^''^foi

Do

(5)

Do

T i m gia trj Idn n h a l cua bieu thiJc P = x^ + y^ + z l

Do

"x = - l

• ] i

X >

s/
up
ro

Da'u b^ng trong (2), (3) tifcJng tfng xay ra

/g

<=> y = - 1 hoSc y = 2; z = - 1 hoac z = 2.

om

C o n g tuTng ve tuTng ve (1), (2), (3) va do x + y + z = 0, nen ta c6: P < 6.

(5)

fa

Nhqn xet: 6 d a y tiif x e [ - 1 ; 2 ] , ta suy ra (1) (diTa v a o dinh l i ve dau ctia tam

Tirdng tiTco:

trong ba so' x, y, z c6 hai so' = 2, mot so' = 0.

2. Bay g i d xet bai toan sau: Cho x, y, z e [0; 2] va x + y + z = 3.

'''
J I

V d i bai toan nay each giai bang each xua't phat tif bat dang thiJe: ,1 ^, ;
(6)

la eha'p nhan dU"Oc. That vay de thay bang each l a m nhif tren: {
, „

xyz = 0

P = 5 o |(2-x)(2-y)(2-z) =0

'

x +y+z =3

[0; 2 ] , ta c6: 0 < x < 2 o x ^ - 2x < 0.

(1)

o

y^-2y<0,
z^-2z<0.

i j N v

hai each nay d a m bao dieu k i e n da'u b^ng xay ra.

w.

ww

G

= o

x+y+z=4

Nhif vay c6 hai each xua't phat tiT ba't dang thuTc ban dau. Chu y la trong ca

Hudng ddn gidi
y, z

xyz

Vay max P = 8 <=> x, y , z thoa man (5). Ta thu lai ket qua tren

(6)c:>P<5- xyz<5

T i m gia t r i Idn nha't cua bieu thtfc P = x^ + y^ + z l

X,

o

o

B a i 5. Cho x, y , z la ba so thifc e [0; 2] va thoa man dieu k i e n x + y + z = 4.

Do

•(x-2)(y-2)(z-2) =0

(2 - x)(2 - y)(2 - z) > 0

ce

V a y max P = 6 o x, y , z thoa man (5).

(**)

T i m gia trj Idn nha't cua bieu thiJc P = x^ + y"^ + z^.

bo

•<» trong ba S O X , y , z CO hai so b i n g - 1 , mot so bkng 2.

ok

dong thcfi c6 dau bkng trong (1). (2), (3) va x + y + z = 0

.c

' D a u b^ng trong (4) xay ra

thtfc bac hai).

(*)

0, y > 0, z > 0 => xyz > 0, nen tif (*) suy ra: P < 8

Da'u bang trong (***) xay ra

J

z^ + z - 2 < 0 .

,

+ 4(x + y + z) - 8 < 0.

+ y + z = 4, nen tif (*) suy ra:

Ta

.

TiTdng tvf, ta c6: y^ + y - 2 < 0,

o

X

(x + y + z ) ^ - ( x ^ + y ^ + z ^ j

xyz - ( 1 6 - P ) + 16 - 8 < ( ) = ^ P < 8 - xyz

HUdng ddn giai

x-2

y, z e |0; 2]

=>xyz-

B a i 4. Cho x, y , z la ba so thyc e [ - 1 ; 2] va thoa man x + y + z = 0.

D a u b^ng trong (1) xay ra

/

=> (x - 2)(y - 2)(z - 2) < 0 => xyz - 2(xy + yz + zx) + 4(x + y + z) - 8 < 0

'^itioi r.few •

NhiT the ta c6 max P = l < = > x = y = z= l .

Dox e [ - l ; 2 ] o - l

X,

iL
ie
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hi
Da
iH
oc
01
/

Cong tCrng ve tirng ve (2), (3), (4) vk c6: P < 1.

v

1. Co the lam each khac nhifsau:

Dafu b^ng trong (3), (4) tiTdng tfng xay ra <=> y = z, z = x.

trong ba so x, y , z e o mot so b^ng 0, mot so b^ng 1, mot so bling 2.

V a y max P = 5 k h i thoa man (7)
•

'

D a u b^ng trong (1), (2), (3) ttfdng tfng xay ra <=>

^x-0

y =0

_x = 2

Ly = 2

z-0

Lz = 2

Tuy nhien each g i a i tir eac bat dang thuTc: 2x - x^ > 0; 2y - y^ > 0; 2z - z^ > 0
la khong chap nhan duTde.
That vay lit (8) ta c6: P - 2(x + y + z) < 0 => P < 6.

''

Tilf (1), (2), (3) c6: x^ + y^ + z^ - 2(x + y + z) < 0.
137

Chuy6n dg BDHS6 Toan gi^ trj I6n nhaft vji gii trj nh6 nhaft - Phan Huy KhSi

2x P = 6 o

Cty TNHH MTV DVVH Khang V\%

=0

p ^ i 7. Cho X , y , z la cac so' thuTc thoa man dieu k i e n x^ + y^ +

T i m gia t r i nho nhat cua bieu thifc P = xyz + 2( 1 + x + y + z + x y + yz + zx).

2y-y2=0

Hudng ddn gidi
X

Tir gia thiet: x^ + y^ + z^= 1

+y+z =3

<1; y <1; z < 1

Tuy nhien he (9) v6 nghiem, v i tif x, y , z e [0; 2] ma tdng x + y + z la so

(1 + x ) ( l + y ) ( l + z) > 0

iL

ie
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nT
hi
Da
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oc
01
/

(1 + x + y + z)

du'dc giai.

Mat

3. Bay g i d lai xet b a i toan: Cho x, y, z e [ - 1 ; 2] va x + y + z = 0.
T i m gia t r i Idn nhat ciia bieu thtfc P = x^ + y^ + z l
Cac ban hay thuT l a i xem tai sao xuat phat tiJf bat dang thiJc:
(x - 2)(y - 2)(z - 2) < 0 thi se khong giai diTcJc!

•S

\•

;

m. \

1

thich hdp sao cho trong bat ding

Do x^ + y^ + z^= 1, nen tiif (3) c6: 1 + x + y + z + x y + yz + zx > 0.

(4)

Cong tilfng ve (2), (4)

(5)

c6 P > 0.
o

Ta
s/
up

P =

/g

HUdng dan gidi

om

y, z e [0; 1], nen ta c6:

ok

.c

(l-x')(l-y) +(l-y')(l-z) + (l-z')(l-x)>0.

(l + x ) ( l + y)(l + z) = 0
l+x+y+z=0

Do x, y, z

(3)

ce

> x^ > x ^ y > y^ > y ^ z > z^ > z l

bo

(2)

y

z

1 + yz • + 1——
+ zx + -1 + xy

j D o x , y , z e [0; l ] , n e n t a c 6 :

o

fa
w.

= z = 1; y = 0

.r-'^-"''' "
^

' d 6 tif (1) suy ra:

V a y max P = 3 o x, y, z thoa man (4).
Nhqn xet: Ba't dang thiJc xua't phat (1), diTa vao dieu k i e n x, y, z e [ 0 ; 1]
dang cua bieu thiJc P dau bai.

X

1+

2x

<
yz x + y +

1+xy

z

2y

y

1 + zx
z

(4)

0; y = z = 1

(2)

khdc de thay: 1 + yz > y + z ( v i n6 ttfdng diTdng v d i (1 - y ) ( l - z) > 0).

jip luan hoaii loan tu'dng tuT c6:

X = y = l;z = 0

'

X < 1; y > 0; z > 0 => X - 1 - yz < 0. L a i do x > 0. nen (2) dung, vay (1) diing.

D a u b^ng trong (3) xay ra o dong thdi c6 dau bkng trong (1), (3)
x=y=z=I

(1)

x(x - 1 - yz) < 0.

ww

(3)

- ^ <
1 + yz
1 + yz + X

at vay (1) o X + xyz + x^ < 2x + 2xyz o x^ - x - xyz < 0

'Mat

( x - U y ^ + z ^ ) + ( x 3 + y ^ + y ^ ) - ( x 2 y + y2z + z 2 x ) < 3 = > P < 3 .

X =

(6)

Hudng ddn gidi

(1)

De t h a y C O o ( x ^ + y ^ + z ^ ) + (x + y + z ) - ( x ^ y + y^z + z ^ x ) < 3.

Nentiif(2)tac6:

x

ro

P = 2 ( x - V y ' ' + z - ' ' l - ( x ^ y + y^z + z^x).

X

'Mi'.

>0.

Bai 8. Cho x, y , z la cac s6 thifc e [0; 1]. T i m gia t r i Idn nha't ciia bieu thtfc

B a i 6. Cho x, y, z la cac so thifc e [0; 1]. T i m gia t r i Idn nhat cua bieu thiJc

=> X

(1) C

Vay m i n P = 0 o x, y, z thoa m a n (6).

luan du'dc max P = a (hoac m i n P = a)!.

[0; 1]

"

<=> trong ba so x, y, z c6 mot s6' b^ng - 1 , 2 so bkng 0.

thi?c P < a (hoac P > a) phai ton tai diem

thuoc m i e n xac dinh cua P de cho dau dang thuTc xay ra. C h i k h i do m d i k c i

G

khac ta l a i c6:

D^u b i n g trong (5) xay ra

Binh ludn: V a n de cot l o i Ih d cho: Phai xuat phat tit bat dang thtfc ban dau

X,

''

= : > l + x + y + z + x y + yz + zx + xyz > 0.

ch^n, nen khong thoa man x + y + z = 3. V i the P < 6. Tuf do bai toan chua

Do

= 1.

x +y+ z

<

2z
x+y+z

)ng tirng v e (3), (4), (5) \h c6: P < 2.

(3)

(4)

(5)

(6)

Ifu hkng trong (6) xay r a o d6ng thcJi c6 dau b^ng trong (3), (4), (5)
o

trong ba so' x, y, z c6 hai so b^ng 1, mot so bang 0.

I^y max P = 2 <:i> x, y, z thoa m a n (7).

(7)


Chuy6n dg BDHSG Join g\i tri Ifln nha't

gi^ trj nh6 nhS't - Phan Huy KhSi

Cty TNHH MTV DWH Khang Vigt

B a i 9. Cho x, y, z 1^ cac so' IhiTc e |0; 11. T i m gia tri Idn nha'l cua bieu thiJc
p = X

+ y^ + z"^ - (xy + yz + zx).

, •

=*x-'+y'+z-Uf^

"

4

2

2

HUdng dan gidi
Dox,y,z6|0;l]=^(l-x)(l-y)(l-z)>0

. - .

=> i - (x + y + z) + (xy + yz + zx) - xyz > 0

'

Do

X,

X

+ y + z - (xy + yz + zx).

'

Vay min P = ^ < = > x = y = z = t = ^ .

(2)

y, z e [0; 1 j => x + y + z > x + y^ +

(3)

=> 1 - xyz > P

'i*--)

L a i do xyz > 0, nen c6: P < 1.

2. C a c bai toan khac suT dung bS't dang thiic Ai tim gia trf Idn nhg't va nho

(4)

nhS't cua h a m s6'

(5)

Trong cac b ^ i todn d muc nay, ta ke't hdp nhieu each khac nhau de sur dung

( l - x ) ( l - y ) ( l - z ) = ()
Dafu b^ng trong (5) xay ra

o

bS't dang thufc trong viec t i m gia tri Idn nha't va nho nha't cua ham so'. C a i dich

y=y

can di den la diTa ve cac ba't d^ng thtfc dang P > a (P < a) trong cac bai toan

z=z

tim gia tri nho nha't cua bieu thtfc (gia t r i Idn nha't cua bieu thuTc P), r o i siJ dung

xyz = 0
<=> trong ba so x, y, z c6 it nha't mot so bang 0, it nhat mot so bang 1 , ^

dinh nghia cua gia t r i Idn nha't va nho nha't da biet. Qua cac bai diTdi day cac

(6)

ban se thay ro them tinh da dang ciia phiTdng phap suf dung bat d i n g thiJc de

x, y, z thda man (6).

giai bai toan t i m gia tri nho nha't va Idn nha't cua chiing ta.

Ta

V a y max P = 1 o

B a i 10. Cho x, y, z, t la cac so thiTc diTdng va thoa man dieu k i e n x + y + z + t =
+ V

T i m gia tri nho nhat cua bieu thtfc P = —;

4

+ Z

4

r

+1
5-.

+ z-^ + f'

L a p luan tiTdng tif ta c6: x'' + 7 / > x z ' + z x '

(3)

— - — + —^-— + —
y +z+1

z+x+1

+ (1 - x ) ( l - y ) ( l - z).

x+y+1

Ttf X, y, z e [0; I ] nen ap dung bat d i n g thufc Co Si ta cd:

y'* + z'* > y z ' + y ' z

(5)

y ' + t' > yt' + y't

(6)

z'' + t^>zl' + zY

w.

bo

(4)

jtoit > ^

gia sur X > y > z.

^^"y"^^^^(i-y)(i-z)(i^y^z)

ce

x'* + t'' > x t ' + t x '

fa

^
=^l>(l-y)(l-z)(l+y+z).

x^.;
(1)
•'Til

(7)

fy = z
<
o
[ y + 2z = 0

0
(8)

Do

X

ffvs

y=z=0

e [0; 1], nen tiT (1) cd: 1 - x > (1 - x ) ( l - y ) ( l - z ) ( l + y + z)

Da'u b^ng trong (8) xay ra <=> dong thdi cd da'u bhng trong (2) - (7)

Ttt (8) suy ra 4(x'* + y"* +

aOi

Da'u bang trong (1) xay r a o l - y = l - z = l + y + z

ww

Cong tCfng vc (2) - (7) ta cd:

P=

Do tinh binh d i n g cua cac bie'n x, y, z nen khong g i a m tdng quat, ta cd the

.c

(1)
(2)

ok

SuT dung ba't dang thi?c: (x - y ) ( x ^ - y ' ' ) > 0
x"* + y" > x y ' + y x '

«

Hudng ddn gidi

om

HU(fng dan gidi

>

Bai 1. Cho x, y, z la cac so thufc e [0; 1]. T i m gia tri Idn nha't ciia b i e u thiJc

/g

x-^ +

.4

s/

4

up

X

ro

•.

b rf>j,t

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01
/

=> 1 - xyz >

Oa'u b^ng trong (9) xay ra <=> da'u bang trong (8) xay ra<=>x = y = z = t = ^ .

(1)

o x = y = z = t = ^ ( d o x + y + z + t = 2)

=>(l-x)(l-y)(l-z) <-i::^.
1+ y + z

z"* +1"*) > x ( x ' ' + y^ + z^ +1) + y ( x U y"* + z^ +1') +

D a u b^ng trong (2) xay ra

z(x-Sy3+z^ + t^) + t ( x V y - V z - U f ' )

y
o

(2)
z=0

x =l


BDHSG Toan g'li trj I6n nhSt

Cty TNHH MTV DWH Khang Vijl

glA trj nh6 nhat - Phan Huy Kh5i

Sur dung: x > y > z > 0, ta c6:

x-^ + l _(x + l)(x^-x^ + x ^ - x ^ + x ^ - x + l)
x'(x + l)
x-\ + l)
1

(3)

-<

1+x+z 1+y+z
z
z
(4)
x + y + 1 1+y+z
1
Viet lai (2) dm dang: (1 - x)(l - y)(l - z) +
1+y+z 1+y+z
Cong ttfng ve' (3), (4), (5) suy ra: P <
hay P < 1.

= x 3 _ x 2 + x - l +i-i- +

(5)
(6)

ro

Ta
s/

up

1. Cacbanhay th^xemcu the (*)la gl? . H ) t,
t>Chung toi difa ra vai trtfcJng hdp cua (*)
:> B«1
Thi du: X = y = z = 1; ho5c x = y = 1; z = 0.
2. Trong bai tren ta da ket hcJp cac phi/dng phap: suT dung tinh bmh ding cua
cdc bien, dung bat ding thuTc Cosi, i/dc liTdng bat ding thtfc... di giki bai
todn.
Bai 2. Cho x, y la cac so thiTc di/dng, thoa man dieu kien xy = 1.
/.

/g

y3

ww

w.

fa

ce

bo

ok

.c

om

Tim gia tri nho nhat cua bieu thtfc P =
+ —.
.
•
1+y 1+X
HUdng ddn giai
"
*' ''^ x'^+y'^ + x'+y^ x^+y'*+(x + y ) ( x 2 - x y + y^)
Vi^t lai P dirdi dang: P =
^
^ =
'-^^
x+l+y+xy
x+y+2
(do xy = 1). R6 rang ta c6: x" + y" > 2xV^ va x^ - xy + y^ > xy. (2)
Dau b^ng trong (2) xay ra o x = y = 1 (do xy = 1).
2x^y^+xy(x + y) „ xy(x + y + 2xy)
,
Tir(l)va (2)suyra: P > ^ l\^l^'P>
^^=xy = I.
x+y+2
x+y+2
Nhrf vay P > 1 va 1dau dingx-*
thtfc xiy ra o x = y = 1.
1
DVay

o xtav =c6l min
=>yP==-X=1 >oP x ==jy ^=r1.+l -^x^(l1+ x) x + 17-^-^
x^(l + x)
Nh4n xet: Xet c&ch giai khdc sau day:
- * ' n ' ' 0'

X0 -x^
( x^1^ 2
f
x+—
+ 2+

—
V x^ ^x +xj

\

\2
1
X + — - 2 x + — + 1.
x+—
Xy
Dat X + - = t, khi do do X > 0 nen
X ta c6 t > 2.
TO (1)Xxet ham so f(t) = t^ - t^ - 2t + 1, vdi t > 2 ' ^'^i ' *^
Ta c6: f'(t) = 3t^ - 2t - 2, va c6 bang bien thien sau:
2
t 1-V7 . l + ^/7
3
3

\i
f'(t)
+
0
0
/
V

Da'u bing trong (6) xay ra o dong thcJi c6 dau bang trong (2), (3), (4) (*)
Vay max P = 1 o x, y, z thoa man (*). t » , ( , , . ! i , t . a > } t ' c >';-ili -

X^

-3 f

X+—

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Chuy6n

A

0
x;
x+—
(1)
+v ^

I'

/
/
/

f(t)

/

Tir do suy ra min P = min f(t) = f(2) = 1
(t > 2). Khi t = 2 thi X + - = 2 o x = 1.
Nhir the min P = 1 o X =X y = 1. Ta thu lai ket qua tren!.
Bii 3. Cho X, y, z Ih cac so thifc di/cJng v^ thoa man dieu kien
12 + —1 + —1 <1 + - + - + - .
X y z
x'' y^
I
Tim gia tri Idn nha't cua bieu thtfc P =
lOx + y + z x + lOy + z• + x• + y + lOz
HU&ng ddn giai
1 1\2 > 0

Ro ring ta c6:
.X 3)
U 3j
1 1 1 2 1 1 1 + 1- > 0 z : > 6 _L _L 1
>4 1 1 1 1-2
- +-+3^ - + - + - 3
x y z;
Ix y z ; (1)
Tiif (1) va theo gia thiet ta c6:

f---T+'


ChuySn dg BDHSG Toan gia trj I6n nhSft vS gia trj nh6 nhift - Phan Huy Kh5i

n
fi
>4
—
+
—
+
—
U
y
zj
Ix—
(I

1

Hudng ddn gidi

0

1

1 1 1 .
-2=>- +- +-—+
X
y z
y zj

+

(2)

—

A p dung ba't d i n g thuTc Cosi cd ban, ta c6: (x + y + z)
, ; Tilf (2), (3) suy ra: X + y + z > 9.
4
4
Ta c6:
lOx + y + z 6x + (4x + y + z)

fl

1

1

—+ —+ -

>9.

V i e t l a i bieu thtfc P diTdi dang:

(3)

1-x

(4)
6x

(5)

4x + y + z

U-x

zJ

^

lOx + y + z

lOx + y + z

^

1
6x

•

1

_

1

^

12x + 4(x + y + z) ~ 12x

4

Tir (5), (6) suy ra:

(6)

4(x + y + z ) '

''V •

1

12x

1

1

4x

4(x + y + z)

lOx + y + z

1

16x

1

-+•

x + y + lOz

16z

16(x + y + z)

1

xj

1

U-z

1

1

+ 3.

S r ^ r . = .:fr

yj

z+x \
2zx

z(y + z)

x(z + x )

y(x + y)

xy

+ —yz +
x(z + x )

up

vyj

I

1" <

z(y + z)

yz
x(z + x )

lvz(y+z)
,

zx
y(x + y)

ok

1

Ix+y

2yz

^

1

+ 2z

+ 3

y)

+3
zx

^

y ( x + y)^
>

yz
Z+X
Vx(z+x) •

7y+z-

zx

+.

Vy(^+y)

(y + z) + (z + x) + (x + y ) ]

R6 rhng ta lai c6:

Vx + y

(2)

> 3 ( x + y + z)

(3)

'f

(10)

16(x + y + z)

bo

l(Dira v ^ o bat d i n g thifc hien nhien (a + b + c)^ > 3(ab + be + ca)). ,.

ce

=

z

+ 2z

(U)

[ T i r (2), (3) suy ra: 3(x + y + z) < 2

fa

+

1

- +- + X
y
z)

y

2

xy

•

_z(y + z)

w.

(4) suy ra: P <

(9)

.c

D a u bkng trong (8), (9) x a y k h i x = y = z.
Cpng tilfng ve' (7), (8), (9) c6: P <

CL-

om

mm m vwi

ro

1

(8)

/g

1

1

^x

Ap dung ba't d i n g thurc Bunhiacopski ta c6:

(7)

16(x + y + z)

1

+ 2y

z)

l,z(y + z)

1

D a u bllng trong (7) x a y ra de thay o x = y = z.
1
1
1
L a p luan tifdng tif, c6:
x + lOy + z 16y 16(x + y + z)

1

x

-2xy

= 3-2

4(x + y + z)

1

1

Ta

4x + y + z ~ 3 x + (x + y + z)

1

^

s/

1

1

U-y

,y+z

1

1-z

+ 2y

1

= 2x

L a i c6:

T i r (2)

n

z

Do x + y + z = 1, nen ta c6:

1
1
4
(Dira vao bat d i n g thiJc Cosi cd ban — + — > - — ^ k h i A > 0, B > 0).
A
B A+ B

^,

1

= 2x

1

1

1-y

y

- +- + -

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1+

Cty TNHH MTV DWH Khang Vi^t

ww

,1
D a u b i n g trong (11) x a y ra o x = y = z = 3. V § y max P = —<::> x = y = z =

; Tir x + y + z = 1, nen tir (4) c6: 2

xy

z(y + z)

+ —yz

x(z + x )

+ —yz

x(z + x )

+ - zx

(x + y + z ) .

y ( x + y)_

+•

zx

>3.

y(x + y)

Nh4n xet: D a y ^ b a i todn tdng hdp sur dung den n h i l u phiTdng phdp:
X u a t phat tir m p t bat d i n g thiirc hien nhien de d i dd'n (1). T i r (1) sur dung

Thay(5)vao(l)vac6:P<0.
D g thay dau b^ng trong (6) x a y ra

d i n g thdrc Cosi c d ban de xdc dinh cac dieu k i e n (2) va (4). R o i l a i ap dun?
o x = y = z = ~ . V a y max P = O o x = y = z = - .
^
3
^
^
3

bat d i n g thuTc Cosi cd ban nhieu Ian de c6 danh gia (11).
B a i 4. Cho x , y, z > 0

thoa m a n dieu k i e n x + y + z = 1.

T i m gii t r i Idn nhat cua bieu thuTc P =

+
1-x

^h4n xet: D a y cung la hhi toan tdng hdp. TriTdc k h i sur dung ba't d i n g thiJc
y
z x
- 2+ - + -

1-y

+\
i-z
U

y

z

Bunhiacopski, ta phai thi/c hipn ph^p nhom cac so hang de diTa P ve dang
(1). T r e n c d sd ( 1 ) , m<3i thiTc hien diTdc phep sur dung bat d i n g thtfc
Bunhiacopski.

•

V

,
- •

145


Chuy6n ai BDHSG Join gia tr| Idn nhSft

gii trj nh6 nhjit - Phan Huy KhSi

Cty TNHH MTV DWH Khang Vi$t

Bai 5. Cho x, y, z > 0 va thoa man dieu kien x + y + z = 2.
Tim gia tri nho nhat cua bieu thtfc

Vay min P = 0 <=> X, y, z thoa man (6).
f^h&n xet: Vice su" dung ba't ding thiJc (2) 1^ dieu phai kheo leo mdi nhan ra.
Bai toan la sur ket hdp nhuan nhuycn giiJa phiTOng phap sur dung mpl bat
dang thtfc da bie't trifdc vati viec ap dung ba't dang thurc Cosi.

HUdng ddn giai
1

DiTa vao gia thiet: x + y + z = 2, ta c6:

,

2(x-^ + y-^ +

Pai 6. Cho x, y, z la cdc so thiTc diTOng va thoa man dieu ki?n

; (*)

.

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Tac6:2P = 2 + ( x ' * + y ^ + z * ) - 2 ( x ' + y ' + z - ^ )

,

•32-3x2=z^ = 16-4y^

) -(x^ + y^ + z^) = x-\ - x) + y ' ( 2 - y) + z3(2 - z )

= x \ y + z) + y \ + x) + z \ + y)

>• •

•

Dg thSy ta CO b^t d i n g thiJc sau:

(1) _ .,
^* '

^

rvs. • ' . u - - ' . .
'
2 16-z^
Tir gia thiet, ta co: y ^ —
- •;

'

TMtvay:

s/

(3)

up

Do (3) dung VI X > 0; y > 0; z > 0. Vay (2) dung.

/g

ro

Dau b^ng trong (2) xay ra o xyz = 0.

bo

+ y + z = 2, n6n ta c6: 2(xy + yz + zx) (x^ + y^ + z^ j < 4 .

=>0V5

^
^
12
•

(1)

(2)

=>xy<V3y^ (doy>0)

(3)
,73

Taeo: xz = ^/3
173

J"

2

X
+

3

,73

(4)

Z'^

yza(y^.z^).

(5)

Tir (3), (4), (5) C6: P <^y^+^(y^+y.^)

fa

r / aaii

(4)

Dau bllng trong (4) xay ra o 2(xy + yz + zx) = x^ + y^ + z^.
Tir(l).(2).(4)suyra: 2(x'+y3+z^)-(x'* + y''+z'*)^2.

v •

=> X < y>y3 (do X > 0, y > 0)

.c

(x + y + z)

ce

X

32-z^

J

Do 5z^ < 16, nen tur (2) suy ra: x^ - 3 y ' < 0 => x^ < 3y^

ok

2(xy + yz + zx) + (x^ +y^ +z^)

om

Ta CO theo bat ddng thtfc Cosi:

Do

=

Ta

< I x^y + xy^ + y'^z + yz'' + z^x + zx'' j + xyz(x + y + z)

2(xy + yz + zx) ^x^ + y^ + z^

2

T a c 6 : x ^ - 3 y ^ = ^ - i i l ^ =
3
4

(2) o x V + xy^ + y^z + yz' + z \ zx'

+

1

L(y^+z^)

w.

(5)

ww

^

X


Doy > z > O i ^ y ' > z ' ^ l ^ _ ^ > z 2 z ^ 5 z ^ < 1 6

x y / x ^ + y ^ ) + y z ( y ^ + z ^ ] + zx(z^+x^)<(xy + yz + z x ) ( x ^ + y ^ + z 2 ) .

j

^ ^^
'^^
' --^^

Tim gia trj nho nhat cua bieu thtfc: P = xy + yz + zx.

^; v /

= x y ( x ^ + y ^ j + y z ( y ^ + z ^ ) + zx(z^+x^).

^'^.^--v:'^-'^

Dau b^ng trong (5) xay ra o dong iWi c6 dau b^ng trong (2) v£l (4)

=>p<

2

z\

2

(6)

xyz = 0
o

Do y^ =

2(xy + yz + zx) = x^ + y^ + z^
x+y+z=2

o trong ba so x, y, z c6 hai so = 1, mot so = 0.

(6)

Tir (*) CO 2P = 2 - r 2 ( x ^ + y^ + z ' ) - ( x ' * + y'^ + z'*)

(7)

Tir(6),(7)suyra:P>0.

,

Dau b^ng trong (8) xay ra o x, y, z thoa man (6)

(8)

P<

^ . nen tuT (6) ta c6:

373 + 1 16-z^

'73+r

z ^ = ^ P < 2 ( 3 7 3 + 1) +

Do(l),nentiir(7)c6: P< 2 ( 3 7 3 + l ) +

r3 + 7 3 ^

(7)

8

3 + 7 3 ^ 1 6 ^ p ^ 3 2 7 3 + 16
5

5

i p a u b^ng trong (8) xay ra o dau b^ng trong (1), (3), (4). (5) dong thdi x5y ra


Chuyen d l BDHSG To^n gi^ tr| I6n nha't

X =

; y^-yz +z ^ ^ l
z^-zx + x ^ ^ l
TUWng tir, ta co:
j>-;
j > - } '
^
y-^ + yz + z
3 z + zx 4- X

3

4^

y = z <=>•
X

„

V a y max P =

•

y = z-

= z

Cty TNHH MTV DWH Khang Vi^t

,

i ' - ^

(9)

a>

1

1

T i r ( l ) . ( 2 ) , ( 4 ) s u y r a : 2P > - ( x + y ) + - ( y +

V5-

32V3 + I 6

...^
i ,4.

,
>' J.. ^

(4)

X+V+7

1

z) + - ( z + x ) ^ P >

.(5)

1

pa'u b^ng trong (5) xay ra o dong thcJi c6 dau bang trong (2), (4)

• ,

, ' ,1

<=> x, y , z thoa man (4).

=y=

4:>X

,

j

Z.

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z =

~1

gia trj nh6 nha't - Phan Huy KhSi

Theo bat d i n g thi?c Cosi cd ban t h i : (x + y + z)

Binh luqn: Day la bai loan long hc(p ve viec suT dung bat d i n g thiJc de t i m gia
tri Idn nha't cua mot bieu thtfc.

«'

X

Do - + - + - = 3 =i> ( X + y + z) > 3.
X
y z

Dira v a o dieu k i e n va dung nhieu bat dang thtfc phu (1), (3), (4), (5) ta c6

Bay gid ket hdp (5), (6) c6 P > 1.

B a i 7. Cho x, y, z 1^ cdc so thi/c di/cfng, thoa man dieu k i e n : - + - + i- = 3 .
•
X
y z

y

z

(6)

Da'u b^ng trong (6) xay ra o x = y = z = 1.

danh gia ve P (xem (6)). lis do suy ra max P.

1 P
—+ —+ — > 9 .

^1

'I

' ^

(7)

DS'u b^ng trong (6) xay ra o dong thdi c6 da'u bang trong (5), (6)
o

T i m gia t r i nho nha't cija bieu thd-c:

X

=y=z= 1

Ta

T6m l a i m i n P = 1 <=> x = y = z = 1.

P =

Blnh luqn: R6 rang day la sir long hrtp cua cac phiTdng phap tiT bien ddi dai so,

z^+zx + x^"

s/

y ^ + y z + z^

up

x ^ + x y + y^

y ^ + y z + z^

z ^ + z x + x^

z^-x'

y^-z3
2

X + x y + y^

2

7

7

y:+yz +z

7

z +zx + x

zr -

—

y-Uz-^
r- +

- T

—

-

X + x y + y^
y ^ + y z + z''
z'^+zx + x^
. ^ . x ^ - x y + y^
,
, y ^ - y z + z^
^
, z ^ - z x + x^
= (x + y ) x^
- ^ + x y + y^
— + i.y + ^)^^r^
y^ + y z + z^7 + (z + x)z^ + zx + x^
Ta nhan tha'y: \ - .
X + xy + y

z + V ( z + x)(z + y)

,

+ y)(x + z) =

3

Tir(l)suyra:

.
X +

(1)

^

(1)
<

V(x + y)(x + z)

(2)
2x + ^ y z

3x^ - 3xy + 3y^ > x^ + xy + y^ o x^ - 2xy + y^ > 0.

V i (3) hien nhien diing nen (2) dung. Da'u bang trong (2) xay ra o x = y .

t

Da'u bang trong (2) xay ra o da'u bang trong (1) xay ra o y = z.
TiTdng liT la c6:

(2)

y + V(y + z ) ( y + x)

2y + %/zx

That vay do x^ + x y + y^ > 0 nen
(2) o

^

,;,

(x + y)(x + z ) > ( x + 7yz)'.

z-Sx-^
+

y + 7(y + z)(y + x)

(X

ww

x-Uy^

x + 7(x + y)(x + z)

A p dung bat dang thiJc Bunhiacopski ta c6:

w.

V a y P = Q. T i r d o suy ra:

z

fa

= (x - y ) + (y - z) + (z - x ) = 0.

y

HUdng ddn giai

ok

2

bo

x'-y»
^

ce

p

.c

Khid6tac6:

X

n

om

x ^ + x y + y^

dung bat dang thuTc cho tru'dc (cac bat dang thtfc (2), (4)), cho den suT dung

Bai 8. Cho x, y , z 1^ cac so thiTc diTdng. T i m gia tri Idn nha't cua bieu thiJc:

/g

X e t dai liTdng: Q =

siJf

bS't dang thiJc Cosi diing de giai bai loan t i m gia t r i nho nha't cua P dat ra.

ro

HUctng ddn giai

z + 7(z + x)(z + y)

(3)
I

2z + ^ x y

•,

(3)

(4)

D a u b i n g trong (3), (4) ti/tJng i^ng xiiy ra <=> z = x; x = y .
14Q


Chiiyen dS BDHSG Toan g\A trj I6n nh^t va glA trj nh6 nhift - Phan Huy KhSi

Cty TNHH MTV DWH Khang Vi^t

Cong tirng ve (2), ( 3 ) , (4) va c 6 : P <
2x + yjyz

1

.P<

1

1
7=

2 + j-y-J^

2+

2y + y/zx

X

y + 7(y + z)(y+ x)

(10)

y/x + yfy +

:

(5)

+-

z

>/7

LSp luan tiTdng tif, la c 6 :

2z + ^fxy

z + yl(z + x)(z + y)

2.j-\j-y-

(11)

\f\+yly+y/z

pa'u bling trong (10), (11) ti/dng tfng xay ra o y = Vzx ; z = ^ x y .
Dau b^ng trong (5) xay ra o (Jong thdi c 6 dau bang trong (2), (3), (4)
f

-5,

;

o x = y = z.

.

Cong tirng ve'(8), (10), (11) ta c 6 : P < 1

(12)
Dau bkng trong (11) xay ra <=> dong thcJi c 6 dau bang trong (8), (10), (11)

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,.,

D a t a = J i . - ; b = / - . I - ; c = j - . J - , thi a > 0, b > 0, c > 0 va abc = 1.
ly

Luc do VP(5) =

1

x = 7yz

vy

1

,

1

2 + a - + 2 + b +2- + c

Theo bat d i n g thiJc Cosi thi: ab + be + ca > 3>/(abc)^ = 3 .

(7)

Tir (6), (7) suy ra: VP(5) < 1

(8)

P < 1.

^:

bo

x + V(x + y)(x + z)

^

<=> x f y + z + 2 , ^ ) <

Ta
s/

,/z^ - z ^ + 3 x y + 6

Ta c 6 : x ' - x^ + 6 - ( 3 x + 3) = x ' - x ^ - 3 x + 3 = x M x ' ^ - l ) - 3 ( x - 1)

= ( x - l ) [ ( x ^ - l ) + (x-^-l) + ( x 2 - l ) ] = ( x - l)^{x^+2x^+3x + 3)

(2)

Do X > 0 => x ' + 2x^ + 3 x > 0, nen tiir (2) suy ra: x^ - x^ + 6 > 3 x + 3

ce
'

ww

X

\ly^-y^ + 3 y z + 6

= (x-l)[xMx^+x +l ) - 3 ] =(x-l)(x^+x^+x^-3)

=> x ' - x^ + 6 + 3 x y > 3(xy + X + 1)

fa
w.

2. Ta con eo each giai khac nhtf sau:

1

Hudng dan giai

1. Bai toan la stf ket hdp giffa vice suf dung cac bat d i n g thtfc Bunhiacopski, bat
dang thiJc Cosi. Do la mot bai toan tdng help.

1

(1)
-x^ +3xy + 6

'

Nhqnxet:

1

P=

ok

V a y m a x P = 1 <=>x = y = z > 0 .

, "

;

Bai 9. Cho x, y, z la ba so di/dng va xyz = 1. Tim gia tri Idn nhaft cua bieu thiJc:

.c

om

Dafu bing trong (8) xay ra <=> a = b = c = 1 o X = y = z > 0.

up

(6) (do abc = 1)

ro

9 + 4(a + b + c)(ab + bc + ca) + (ab + bc + ca)

<=> Vx (7y + Vz) ^

;

thtfcay!.

/g

9 + 4(a + b + c)(ab + be + ca) + 3

That vay (8) <=>x^/x + Xyjy +

x = y = z>0. /.,

(11). Tuy nhien no kh6ng tif nhien d cho vi sao lai xuat phat tijf cac ba't ding

. I

8 + 4(a + b + c) + 2(ab + bc + ca) + abc

Ta se chtfug minh \d\, y, z > 0, thi:

o

Binh luan: Cach giai nay chi difa vao cac bat dang thuTc biet triTdc (8), (10),

12 + 4(a + b + c) + (ab + be + ca)

'

<:> • y = yfzx

Vay max P = 1 o x = y = z > 0. Ta thu lai ket qua tren.

(2 + a)(2 + b)(2 + c)

/

t

z = 7xy

(2 + b)(2 + c) + (2 + c)(2 + a) + (2 + a)(2 + b)

' h !

V-

yj\

(8)

yfx+yjy+ylz

< xVx + Vx•^/(x + y)(x + z)

+y ) ( ^ + v

VI (9) dung, vay (8) dung. Dau b^ng trong (8) xay ra <=> x = yjyz .

J

<

^ x 5 - x 2 + 6 + 3xy

^

- •

(3)

V3(xy + x + l )

Dau bkng trong ( 3 ) xay ra o dau b^ng trong (2) xay ra <^ x = 1.

/'

TiWng tif, ta c 6 :

'

x^ + xz + x y + yz r ^ ^ ' ' - ^ *• V '

<=> 2x7yz < x^ + y z o 0 < ( x - 7 y z ) ^ .

=>

(9)

< —
y^ + 3 y z + 6 V3(yz + y + l )
1

I

7z^-z2+3xy + 6

V3(zx + z + l )

Da'u b^ng trong (4), (5) xay ra tifdng i?ng <=> y = 1, z = 1.
Cong tfifng ve ( 3 ) , (4), ( 5 ) va c 6

i ^ L

(4)
(5)


Chuyen 66 BDHSG Toan gii tr| Idn nha't va gia tr| nh6 nhS'l - Phan Huy KhSi

y/3 y yjxy + X + 1

(6)

yjyz + y + 1

D a u b^ng trong (6) xay ra o

Cty TNHH MTV DWH KhanQ Vigt

Vzx + z + 1^

dong thcfi c6 dau b^ng trong (3), (4), (5)

<z> X = y = z = 1.

9-(x^+y^+z^

v a y P c6 dang: P = \/\ + ylz -

=:>2P = x 2 + y 2 + 9 + 2 ( 7 ^ + 7 y + N ^ ) - 9 .

,

x^ +

A p dung bat d i n g thufc Bunhiacopski, ta c6:

^ xy +

1
X

(1 + 1 + 1)> ,

+
-+
zx + z +1 ^
; ^ x y + x + l Vyz + y + 1 N/ZX

-+yz + y + 1

+1

1
^xy +

\2

1

1
X

+1

'

1

^yz + y +1

-Jzx + z + l

1

xy +

\

v'-,

D a u bling trong (7) xay ra o

1
+1

1

+ •yz + y + 1

Ta
zx + z + l

.:^:t;vv^ i

xy

X

xy + x + 1

'

1 + xy + x

'

s/

-

om

xy + x + 1

.c

,

dong thcti c6 da'u bkng trong (6), (7)

ce

V a y max P = l o x = y = z = l .

(3), (4), (5)) va bat dang thiJc

w.

thuTc

fa

Binh luqn: B a i nay la sir ket hdp giffa viec suT dung mot bat dang thtfc da bici
trirdc (trong bai nay la cac bat dang

(4)
;

= Vz

(5)

ww

Bunhiacopski. C a i kho c( day la phat hien ra (3), (4), (5).
NhiT vay mot Ian nffa cac ban thay de giai mot bai toan t i m gia t r i Idn nhiil
va nho nha't nhieu k h i phai kheo 16o ke't hdp nhieu phU'Ong phap khac nhau,
chur khong ddn thuan chi dung mot phiTdng phap la d u ! .
B a i 10. Cho x > 0, y > 0, z > 0 va thoa man dieu k i e n : x + y + z = 3.

x = y = z = 1.

phap them bdt hang tuT k h i dung ba't dang thiJc Cosi.

-) /; .

Bjki 11. Cho x, y, z la cac so thirc khong a m va thoa m a n dieu k i e n :
+ y^ +

= 3. T i m gid t r i Idn nha't cua bieu thufc:

p =

X

+2y + 3

y'^+2z + 3

z +2x + 3
HUdng ddn gidi

+ 1 > 2x; y H 1 > 2y; z^ + 1 > 2z, nen ta cd

Vi
p

^

—

^

—

+

2(x + y + l)

—

I

—

+

2(y + z + l )

—

f

—

1,, V-

.

2(z + x + l )

Da'u b^ng trong (1) xay r a o x = y = z = l (khi do thoa m a n

+ y^ + z^ = 3).

Ta cd:
X+y+1

y+Z+1

Z+X+1

1

T i m gia t r i nho nha't cua bieu thtfc: P = >/x + 7y + Vz - (xy + yz + z x ) .
= 3-

HUdng ddn gidi
^
,
(x + y + z ) ^ - ( x ^ + y ^ + z ^ )
9-x^-y^-z^
Ta c6: xy + yz + zx =
^
'- =
.

,

Nhqn xet: Mau chot cua bai toan la du'a P ve dang (1), sau do suf dung phiTdng

bo

<::>x = y = z = l .

z^ + %/z + >/z > 3z

v a y m i n P = 0<=>x = y = z = l .

(10)

ok

Da'u b^ng trong (10) xay ra o

y

x + l + xy

^ 1 + x + xy _^

Thay (9) v a o ( 8 ) va c6: P < 1.

(3)

Tilf(l), ( 5 ) c 6 : P > 0 .

up

+ •

+ ^^^y

X = y = z = 1 (chii y luc do thoa man: x + y + z = 3).

o

(8)

ro

yz + y + 1

(2)

Da'u bkng trong (5) xay ra

/g

-+

xy + x + 1

1

3x

y, z > 0).

X,

Dau bang trong (6) xay ra o

1

>

x^ + y^ + z^+ 2(Vx+7y+%/z)>3(x + y + z) = 9.

= y = z = 1 (do xyz = 1).

i
+
^

+
^
.
xy + x + 1 yz + y + I zx + z + i

1

+ N/X

()

Cpng turng ve (1), (2), (3) ta c6:

(7)

.

1; y = 1; z = 1 (do

<:> X =

zx + z + l)

V i x y z = l,nen:
1

y' + ^

(1)

Da'u bang trong (2), (3), (4) tiTdng iJng xay ra k h i x^ = Vx ; y^ =

1

xy + x + 1 = yz + y + 1 = zx + zx +1

<=> X

Tiif (6), (7), ta c6 P <

X

+ Z +

A p dung bat d i n g thiJc Cosi, ta c6

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1

N/X

)

T f i f d ) , (2) s u y r a :

y +1
x+y +1
y+1
^x+y+1

1--

z +1
y+

Z +

1

+ 1-

z+1

x+1

y+z+1

z+x+1

x+1

z+x+ U


CtyTNHH MTV DWH Khang Vi^t

Chuy6n ai BDHSG To^n gia trj lOn nhft va gia tri nh6 nh^t - Phan Huy Khii

p

2 2

Ttf (2) dung suy ra (1) diing. Nhan xet dtfdc chiJng minh.
R6 rang trong ba so x + y^, y + z^, z + x^ c6 it nha't hai so hang cCing dau. Vi
thd' c6 the gia sur (ma khong lam ma't tinh tong quat) (x + y^ )(y + ) > 0.
Ttf (2) v£l theo (1), ta c6: ^ l + x + y^ + -y/l + y + z^ ^ 1 + ^ l + x + y^ +y + z^
TCf do suy ra:
Jl + x + y^ +-y/l + y + z^ + Vl + z + x^ > l + 7 l + x + y^ + y + z^ +>/l + z + x^
=> P > >/I + ( N / I - Z + Z2)77 yjiyfU^f+x^
^
^'
(3)
(Chu y: do X + y = - z V I x + y + z = 0).
Ap dung nhan xet sau day: Vdi moi so thi/c a, b, c, d ta c6:
V a ^ + b ^ + V c ^ + d ^ >yl(a + cf +(h + df . > i
(*)
Tiydo ta c6:

^ y+1
z+1

x+1^
x + y + 1 y + z + 1 z + x + 1^

(x + l)2
2 2L(y + l)(x + y + l) (z + l)(y + z + l) (x + l)(z + x + l)
Theo bat ding thtfc Svac-xd, ta c6: i , ; , ^ .
^.
(y + l)2
,
(z + lf
(x + lf
'
-+•
(y + l)(x + y + l) (z + l)(y + z + l) (x + l)(z + x + l)
(y + l + z + l + x + 1)^
(4)
(y + l)(x + y +1) + (z + l)(y + z +1) + (x + l)(z + x +1)',
De y r^ng do: x^ + y^ + z^ = 3, nen ta c6:
(y + l)(x + y + 1) + (z + l)(y + z + 1) + (x + l)(z + x + 1) . t '"^ ,
= 3(x
2 + y + z) + xy + yz + zx + x^ + y^ + z^ + 3
J,
= - (x^ + y^ + z^ + 9 + 6x + 6y + 6z + 2xy + 2yz + 2zx) = - (x + y + z + 3)^(5)
,

(z + lf

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(y + lf

+

•p'-i it i .

Ta

i
> 2.
(x + l)(z + x + l)

bo

ok

.c

(y + l)(x + y + l) (z + l)(y + z + l)
Da'u b^ng trong (6) xay ra o X = y = z = 1.
Tir(3), (5)suyra:P < ^
yWfiS

Dau b^ng trong (7) xay ra o dong thdi c6 da'u b^ng trong (3), (6)
<=> X = y = z = 1

up

+

ro

+

Vay max P = •^
ce

•

^ ,.

ww

w.

fa

Binh luan: R6 rang day la bai toan ket hdp bat ding thiJc Cosi, ba't ding thiJc
Svac-xcJ cdng vdi nhieu phep bien ddi dai so.
Bai 12. Cho x, y, z la cac so thifc e [-1; 1] va thoa man dieu kien x + y + z = 0.
Tim gia tri nho nha't cua bieu thtfc
P = yjl + x + y^ +yjl + y + z^ +Vl + z + x^ .

,

/(7l-z + z^f+y2

(x + 1)^

. ^

,

' , ,

Hudngddngiai
,
Ta c6 nhan x6t sau day: Neu ab> 0; a, b v^ a + b deu > -1. Khi d6
Vl + a + Vl + b > l + Vl + a + b
That vay sau khi binh phiTdng ca hai ve cua (1), ta c6:
(1) <=>2 + a + b + 2V(l + a)(l + b ) > 2 + a + b + 2Vl+a + b
<=>(!+a)(l+b)> 1 + a + b o a b > 0 .

s/

(z + 1)^

/g

Tir(4), (5)suyra:

(y + 1)^

om

TV/.t^

(2)

+,J{J^f

+ J^2 > ^ ( 7 i _ z + z2 + Vf+z f + (x + y)^ .

(4)

Tir(3), ( 4 ) v a d o x + y + z = 0,nenc6: P > 1 + v ( V l ^ z + z^ + VT+z) +z^ (5)
Talaico: ( V l - z + z^+VrTz) +z^

«

= l - z + z^ + l + z + 2 V ( l - z + z^)(l + z) + z^
= 2 + 2 z ^ + 2 V l - z ^ + z ^ + z ^ =2z2+2 + 2Vl

^» •<
-jjj^iS

+ z^ .

(6)

Ta se chtfng minh: 2z^ + 2 + 2Vl + z^ > 4.
'"

That vay (7) o z^ +1 + Vl + z^ ^ 2 o Vl + z^ > 1 - z ^
"
Do 1 - z^ > 0 nen (7) » 1 + z' > 1 - 2z^ + z" o + 2z^ - z* ^ 0

(7)

o z K z + 2-z2)>0.

(8)

Do z e [-1; 1] => z - z^ ^ -2 z + 2 - z^ > 0, vay (8) dung => (7) dung.
Tilf (5), (7) suy ra P > 3.
(9)
Da'u b^ng trong (9) xay ra de tha'y o x = y = z = 0 (cac ban tif nghi$m lai).
Vay minP = 3«x = y = z = 0.
^inh luan: Bki toan thUc stf la bai todn tong hdp. Ma'u chot Ik dxia. vao bat ding
thtfc (1), de suy ra danh gid (3). Lai drfa vao ba't ding thuTc hien nhien (*), de
It c6 danh gid (5). Sau do lai difa v^o bat ding thuTc (7) de c6 danh gia (9).
^^i 13. Cho X, y, z Ik cde so thiTc diTcfng va thoa man dieu kidn: xyz = 8.
Tim gid tri nh«5 nha't cua bieu thtfc:
iit..
' '. V
,55


ChuySn dg BDHSG ToAn giA tri Idn nhSt va gi^ trj nh6 nhaTt - Phan Huy KhSi

Cty TMHH MTV DWH Khang Vi$t

pai 14. Cho x, y, z, t la cac so diTdng va thoa man dieu kien: xyzt = 1.

V ( i + x-^)(i + y ' )

V{i + y )(i +

^

Tim gia tri nho nhat cua bieu thiJc:

z 3 ) ( i + x-^)

1
HUdngdangidi

. v i >

^~

1

Ap dung (1) ta c6:

(1 + af

^

4z'

4

4z2

""77::

i
(1)

7:;- ; Y

l + 2b + b^+2a + 4ab + a^+2ab(a + b) + a^b^

l + ab(a2+b^)>2ab + a V . ,

(2)
liiv

:

(3)

Dau b^ng trong (3) xay ra o a = b.

s/

(2 + 4a)(2 + 4b)

(2 + 4b)(2 + 4c)

(2 + 4c)(2 + 4a)
c

( i + 2b)(l + 2c)

( l + 2c)(l + 2a).

Vay 2a^b^ + 1 > 2ab + a ' b l

l + 2(a + b + c) + 4(ab + bc + ca) + 8abc

Dau bkng trong (4) xay ra o ab = 1.

*t
"

Da'u bkng trong (1) xay ra o dong thdi c6 dau bang trong (3), (4) , j ^ i

'

' *'

Da'u b^ng trong (1) xay ra

ww

w.

Ta c6: a + b + c > 3^/abc = 3,

-

Tir (2), (3), (4) suy ra (2) dung, ttfc Ih (1) dung

1 + xy

fa

ce

•

(4)

,

Ap dung nhan xet (1), ta CO: P > — +

bo

a + b + c + 2(ab + bc + ca)

it' •

o a = b = 1.

ok

( l + 2a)(l + 2 b ) ( l + 2c)

ab + be + ca > 3^(abc)^ = 3,

X6t hieu: (2a^b2 +1) -(2ab + a^b^) = a^b^ - 2 a b +1 = (ab - 1 f .

.c

a(l + 2c) + b ( l + 2a) + c ( l + 2b)

/g

. ( l + 2a)(l + 2b)

b

om

a

up

4c

ro

16b

^'p^-^

i

= ^ l + ab(a^+b2)> 2 a V + l

abc = 1.

16a

P>4

i

i

D o a > 0 , b > 0 n e n a ^ + b^>2ab

Luc do tiif (3) suy ra.

hay P > 4

1_

(1 + b)2 " 1 + ab

o

4

Khi do do X, y, z > 0 va xyz = 8 => a > 0, b > 0, c > 0

P>

.

>

<

z^

4

1 1 ,.

<:> 2 + 2(a + b) + a^ +b^ +2ab + 2ab(a + b) + ab(a^ + b ^ ) >

" ( 2 + x2)(2 + y ^ ) ^ ( 2 + y2)(2 + z 2 ) ^ 2 f z 2 ) ( 2 + x 2 ) "
x^

1

«

Ta

' ,

4y2

^

T h a t v a y : ( l ) o (2 + 2a + 2b + a 2 + b 2 ) ( l + a b ) > ( l + 2a + a2)(l + 2b + b2)

'

V4(l + x-^)4(l + y ^ ) ^ V 4 ( i + y ' ) 4 ( l + z ' ) ^ > / 4 ( l + z 3 ) 4 ( l + x 3 )
4x^

HUdngddngidi

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1

4y^

f

-

Ta CO nhan xet sau: V d i moi so dating a, b ta co:

Do (2) dung nen (1) dung.

^'

l.t ;

4

That vay (1) o 4 + 4 a ' < a'* + 4a^ + 4 o a" - 4a'+ 4a^ > 0 o a\ - 2)^ > 0 ,

4x^

1

(1 + x ) ' ^ ( 1 + y ) ' " " ( I + Z ) ' ' ' ( l + t)2 •

N M n x6t: Va thi 4 ( l + a"*) <(a^ + 2)^.

p

1

rr

=> a + b + c + 2(ab + be + ca)> 9 = 1 + Babe

^

1

1

Tac6:

x =y=l

z=t=1

1 + zt + l + xy

+
1 + xy

=> (1 + Babe) + [2(a + b + c) + 4(ab + be + ca)]

o

=
1 + zt

( l + xy)(l + zt)

—!—.

'

"

- V

<=>x = y = z = t = l .

? i

2 + xy + zt

,
.
= 1 (do xyzl = 1).
1 + xy + zt + xyzt

Vay tir(4)e6: P > 1.
<3[(a + b + c) + 2(ab + bc + ea)].
Tir(4),(5)suyra:P > | .
thay dau b^ng trong (6) xay ra
<=>a = b = e = l < = > x = y = z = 2.
4

V^y min P = — C:>x = y = z = 2.

"

(5)

(4)

'\.,.X.

1 + zt

(3)
i gfi i ' '

Dau bang trong (5) xay ra <=> x = y = z = t = 1.
Vay min P = l o x = y = z = t = l .
^hgn xet: Quan trong nhat Ih biet diing bat d i n g thiJc phu (1).

fi^i 15. Cho X, y, z la ede so' thifc diTdng va thoa man dieu kien:
x y z
x + y+ z > - + —+ —.
y

2

\ . ' ; f >

•

.157


Cty TNHH MTV DVVH Khang Vigt

Chuy6n ai BDHSG Toan gia trj Idn nhift \ik g\i trj nh6 nha't - Phan Huy Khii

Tim gia tri nho nha't cua bieu thiJc: P =

+
x +1

+
y +

1

3(xy + yz + zx) + 3(x + y + z) = 2(xy + yz + zx) + (xy + yz + zx) + 3(x + y + z).

.
z+1

Do (5) suy ra: 3(xy + yz + zx) + 3(x + y + z) > 2(xy + yz + zx) + 4(x + y + z).
3
Thay (10), (11) v^o (9) suy ra VP(2) < - , tCr d6 theo (2) c6:

• r>i th

Hiidng dan giai
V i e t l a i P d i r d i dang:

^
,
Taco:

+l

y

1

^x + 1- + y

z+1

(y + l)(z +1) +

1
1

1
+
+
x+1 • y+1 •z+1

(X

+

1

! 1 -

1 + z- + 1^

+ l)(z +1) +

(x + l)(y + l)(z + l )

,

{

,

V

y

z

xy

X

yz

zx

,x J

• >, i /

2
,

DSu bkng trong (13) xdy ra

(2)

O dong thcJi c6 dau bang trong cac bat ding thtfc (5), (6), (7), (8), (11)

z

o x =y= z=l,

(3)

Turgia thiettaco: x + y + z > - + - + y z x
X

y z x^ y^ z^
(x + y + z)^
nhiTng: - + i + - = — + 1- + — >1
1 i _

z+1

T t f ( l ) , (12) ta c6:P > - .

.

xyz + (xy + yz + zx) + (x + y + z) + l
X

y+1

Dau bang trong (12) xay ra<=>x = y = z = l .

+ l)(y +1)

(X

(xy + yz + zx) + 2(x + y + z) + 3
' ',

x+1

(1)

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P= 1-x+1

Vay min P = - o x = y = z = l .

(4)

Binh ludn: Ro rang day la bai todn long hdp, sijr dung rS't nhieu bat d i n g thuTc

xy + yz + zx

Ta

phu, bat dang thuTc Cosi, bat ding thiJc Svac-xd,... d^ gidi bai loan.

(theo baft d i n g thtfc Svac-xcJ)

=> xy + yz + zx >

X

+ y + z.

(x + y + z)

X+ y +z
-=>1>
xy + yz + zx
xy + yz + zx

s/

+ y + z>

up

X

ro

Tir (3), (4) suy ra:

•

.

X

1

1 -^z
X
y
x y z 1
2- + - + - 2- + + y + z > - + i - + - = _ 2 - + - +—
3l Z
X;
3k X
y)
z>
y z X 3l y

^ = y

y

^>3

ce

y

X

— = •
, nen tUOng i\i ta c6:
yz
^xyz
2^ + i

ww

^/xyz

'4xyz

Tir (6), (7) suy ra: x + y + z > ^ + y + z _^
^/xyz

^^

D^u b^ng trong (8) xay r a o x = y = z = l .
Tac6: VP(2) = 3 2

+V y 3 + ( z

•+

+ x)^ ' 'yz''+(x + y)^ '

Hiidngddngidi

' ^' '

Ta c6 nhan xet sau: vdi moi x, y, z la cac s6'thifc dufOng, ta c 6 : '

' '

2

V x ^ + ( y + z)"'
Thatvay:(1)

x^+y^+z^
»
x ^ + ( y + z)^

(x^+y^+z^)

w.

fa

VI:

X

bo

ok

Ta lai co:

lx^+(y + z)-'

.c

.

om

Dau bing trong (5) xay r a o x = y = z = l .
r,,

P=

/g

(5)

Bai 16. Cho x, y, z la cdc so thiTc diTdng. Tim gia tri nho nha't cua bieu thiJc:

(xy + yz + zx) + 2(x + y + z) + 3

o x^ [x^ + 2x2 (y2 +

) + (y^ + z^) J > x U x ^ x + z)^
2

o 2 x 2 + ( y 2 + z 2 ) + (y2+z2)

>x(y

+ z)^

.ai?

-

-

Theo bat d i n g thtfc Co Si, ta c6:
2x2 + 2 x H y ' + z ' ) + ( y ' + z2 f > 2^/2x2 ( y ^ + z ^ f .

2 3 xyz + (xy + yz + zx) + (x + y + z) + 1
_ 3

2(xy + yz + zx) + 4(x + y + z) + 6

2 ' 3xyz + 3(xy + yz + zx) + 3(x + y + z) + 3
Tilf (8) suy ra: 3xyz + 3 > 6,

Ro rang: 2(y2 + z^) > (y + z)^.

?,

Tir (3), (4) suy ra: 2x2(y2+/.-) + (y2+z2) > Jx2(y + z)^ = x ( y + z)^
Tir (4) suy ra (2) dung, vay (1) dung. Da'u bang trong (1) xay ra <^ y = z.
159


Chuy6n Oi BDHSG Toan gii trj I6n nha't

gia tri nho nhai - Phan Huy Kh5i

Cty TIMHH MTV DWH Khang Viet

p a i 18. Cho x, y, z la cac so' thi/c Ihoa man: x"* + y^ + z/* = 3.
Tiftlng t i r < l ) la c6:

——=
\ + (z + x)

T-~
X'

1
+ y" + z

7 •

'^*>

T i m gia trj \dn nha't cua bieu thiJc: P = x ' y ' + y V.' + z \
HUt'fng ddn

z

z

z -(-(x + yr

gidi

Theo bat dang thiJc Cosi (cho 4 so'), ta c6:

x'^+y

+z+ y ' + 1 + 1 > 4xy

"

'

=^ xV (x^ + y"* + 2 ) > 4 x V ' .

(1)

iL
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oc
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/

Dau bang trong (6), (7) tiTdng uTng xiiy ra c=> /, = x; x = y.
Da'u bang trong (1) xay ra <=> |x| = |y .

Cong lirng ve (1), (5). (6) va co: P > 1.
Da'u biing trong (7) xay ra <=> dong ihdi c6 dau bang trong (1), (5), (6)

O X = y = Z > {)

,

• .iU'-!^

V a y min P = l o x = y = z > 0 .
^_

^ ^ „

l+x'+y;

A p dung ba't dang thii'c Bunhiacopski, ta c6:

i

' •

^

x(l + y^+z^) + y ( l + z?+x^) + z(l + x ' + y ^ )

+

;

,

I

+

1 + z^+x^

^

l+ x'+y'

zy.

om

(1)

ok

.c

Tiif (1) va do X + y + z = 1, la c6:

ce

Dat Q = x ( l + y^ + z^) + y ( l + z^ + x ^ ) + z ( l + x^ + y ^ )

bo

" x ( l + y - + z ^ ) + y ( l + z ^ + x^) + z ( l + x ^ + y ^ ) '

ww

w.

= 1 + xy(x + y ) + yz(z + y) + zx(z + x)

fa

+ y + z) + xy(x + y) + yz(z + y) + zx(z + x)

'••'••±: 1 + x^(y + z) + y^(z + x) + z^(x + y).

Co the thay rang: x^(y + z) + y'(z + x) + z'(x + y ) ^ ^
(ban doc thilr nghiem lai xcm!)

i

- f"''

/g

1 + y^+z^

= (X

Da'u bc^ng trong (4) xiiy ra o

-''''hi^^

HUl'fng ddn gidi

>(x + y +

(3)

,

(4)

dong thcJi co da'u bang trong (1), ( 2 ) , (3)

Dat a = x ' ; b = y*; c = /*, ta co a > 0; b > 0; c > 0 va a + b + c = x" + y ' + z' = 3.
Khi do VP(4) = ab(a + b + 2) + bc(b + c + 2) + ca(c + a + 2)
= ab(5 - c) + bc(5 - a) + ca(5 - b)

(do a + b + c = 3)

= 5(ab + be + ca) - 3abc.

(5)

R6 rang ta co bat dang thii'c sau: (a + b - c)(b + c - a)(a + c - b) < abc
=> (3 - 2c)(3 - 2a)(3 - 2b) < abc

.:

(6)

•" '

=^ 27 - 18(a + b + c) + 12(ab + be + ca) - 8abe < abc
=> 12(ab + bc + c a ) - 9 a b c < 2 7

4(ab + be + ca) - 3abc < 9.

Dau bang trong (6) xay ra <:>a = b = e.

Ta co: ab + be + ea <

(a + b + e)^

Dau b^ng trong (6) xay ra o

(6)

-iv::::',.:

= 3.

(7)

a = b = e.

Tur (6), (7) co: 5(ab + be + ca) - 3abc < 12.

(8)

Tir (4), (5), (8) suy ra: 4P < 12 =:> P < 3.

V*

o a
<=>

V a y max P = 3

o

'

'

dong thcJi co da'u b^ng trong (6), (7), (8)
= b = c= 1
X

TCr (2), (3) suy ra: P > - . V a y m i n P = - .
Gia trj nho nha't dat diTdc k h i x = y = ^ ; z = 0.

izUxUl).

o

Da'u bkng trong (9) xay ra o
Tird6c6:Q < - .

|y

4P < x^-* (x^ + y-* + 2 ) + y^z"* (y'* + y' + 2 ) + y.\'

-.

Ta

J :^^..'K^\..J.t^l^

1

s/

+

up

I

ro

+

zV(z-' + x ^ + 2 ) > 4 z V .

Cong tirng ve (1), ( 2 ) , (3) va c6:

T i m gia tri nho nha't cua bieu thii'c:
2.

(2)

Da'u bang trong ( 2 ) , (3) tu'dng uTng xay ra o

B a i 17. Cho x, y, z > 0 va ihoa man dicu k i c n x + y + z = 1.

p=

TiTdng tiT, ta c6: x''z''(y'* + /* +2)> 4y'*z\

= 1.

=

1.

(9)


Cty T N H H M T V D W H

ChuySn 6i B D H S G Todn gia trj I6n nh9't vS gia tr| nh6 nhjl - Phan Huy Kh^i

Nhdnxet:
,
>.,1. Day la bai loan tdng hdp suf dung nhieu bal dang thtfc. ,|„ ,^ ,
2. (6) ChuTng minh nhir sau:
Ta chi xet khi (a + b - c)(b + c - a)(c + a - b) >()

^4 •( y'i:

abc > 0).

(vi

y ' 2

Dau bang trong (5) x a y ra <=>

Chi C O hai kha nang:

Vay max P =

a) Co hai Ihiifa so am, mot ihuTa so dufdng. Gia suf

^

^ =2

'Hfitr'!,

= 2;

'

y = 1.-

{

ic)

,;,|.:..:f

(6)


X=l;y = 2

<z>

=

2;y

=

r

*

iL
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x

•a + b - c < 0 = > 2 b < 0 = : > b < 0 ( v 6 1 i V I bSO)

X = 1; y = 2
_x

Khang V i j t

•'-'OJ

^

^

2 Do vai tro bmh dang giffa x. y, z nen ta c6 the gia suT y la so hang giffa trong

<

t'
[b + c - a < 0 => khong xay ra irtfdng hdp n i y .
(. b) Cabathirasodifting
^ / v > r-, H^'

< ".mMxy %tc

ba so X, y. z. Khi do ta c6:
Q_x__^y_^z_^j^x__^z_

Dat X = a + b - c; Y = b + c - a; Z = c + a - b =^ X > 0; Y > 0; Z > 0

y

X ~

>

i S!)x,.~H

7.

That vay (7)

Ta c6 theo ba'l ding thiJc Cosi: (X + Y)(Y + Z)(Z + X) > 8XYZ
=>8abc>8(a + b - c ) ( b + c - a ) ( c + a - b )

z

' - * "t
- ' !
Mi',m^^:Mi 'Adi'h^.
iA x ,V;

' ••

•>

sftVidflu! .

XJ^^^w*!'

•

'

:

v'- '

oiiy,i-^>()^^y-y'^y^-'-^>o^<^-yHy-^)>o.
/
y
yz
yz

=> (6) diTcJc chuTng minh!.
Bai 19.

(8)

Do y la so hang giOfa nen (8) dung vay (7) diing

X, y

V

Z

y

z

X

s/

X

om

/g

ro

up

2. Cho X, y, z 6 11; 2]. Tim gia tri Idn nhat cua bieu thiJc P = - + i - + - .

T i r ( l ) s u y ra:
<

0

^

vy/

+ 1<0=>

\

ok

w.

^

X

ww

--2

X

bo

.

ce

V i = i < » x = l;y = 2 c 6 n - = 2 o
ly
2
y

Cong lifng ve(2), (3) va c6:

X y
—+ —
ly
X,

<-

X

X

(2)
(3)

+ 1<
y

(4)

' *

.^.tj,j

1

I

"i /ji?n

Do do max Q = — .

^
^
-Ji^i. -y.*. -xj.

2

:'^.olw:iAvu."::.

j

Hid'ing dan gidi

I ir

2

Do - + y > 0, nen tir (4) c6: P = - + ^ < I.
V X
y
X 2

;

Tim gia tri \dn nhii't ciia bieu thufc: P = x' + xy + y'.

Si

2 vyy

,^ Do vai tro binh dang giffa x, y nen ta cung c6:

Mat khiic chang han khi: x = 1; y = 1; z = 2 thi O = —.

Bai 20. Cht) x, y la cac so thifc ihoa man: |6x -3y| < 9 va |x - 3y| < 1 .

.c

=>-<-<2

2 y

fa

1. D o x . y e [1;2]

X z 5
7
Theo phan 1. ta c6 - + - < - , vay tu'(7) suy ra: Q < - Vx, y, z e [ I ; 2].
. . z X 2
2

Ta

1. Cho X, y 6 [ 1; 21. Tim gid Iri Idn nha't cua bieu thtfc P = - + ^ .

(5)
'

TiTgia thictlaco: |6x-3y|<9; | x - 3 y [ < I ,

(1)

2x--y <3: 2 x - 6 y < 2 .

(2)

Sijf dung tinh chat sau day ciia gia tri luyct doi:
A + B| < |A| + |B| va da'u b;lng xay ra o A B > 0.
Tif(l),(2)lac6:

5x =
5y = ( 2 x - y ) f ( 6 y - 2 x )
Tir do suy ra:

2 x - y + 2 x - 6 y <3 + 2 = 5.,, ,,

< 2,
<1.

.

J

... . . / O !

,

,
(3)
(4)

Da'u bhng trong (3) xay ra <^ (6x - 3y)(3y - x) > 0.
163


Chuyen dg BDHSG Toan gia Iri I6n nhait vi g\i tr| nh6 nha't - Phan Huy Kh^i

Da'u bang trong (4) xiiy ra «
Ta

CO

Cty TNHH MTV DVVH Khang Vift

(2x - y)(6y - 2x) > 0.

P = X ' + xy + y ' < X ' + |xy| + y ' .

'^

pai 22. Cho x, y, z la ba so' dU'i^ng ihoa man dieu kiOn xyz = 1.

'

(5)

Thay (3), (4) viio (5) va c6: P < 4 + 2 + 1 = 7.

''

'

T i m gia tri nho nha't ciia bicu thi?c

(6)

P=

Da'u bang trong (6) xay ra <=> dong th(1i c6 da'u bang trong (3), (4)

!
x"*(x + y)

1

r

1

x = 2;y = l

I V i c t lai P duTli dang: P -

>si M> •

iiA/-?.. .^r/ih jf|>r;x ,r,Jr

_x = - 2 ; y = - l .

= 1.

v

v

«

,

T\m gia Iri nho nha't cua bicu thiJc: P = x(y + /.) + /(x + y).

V i c t lai P dvtiVi diing: P = (xy + yz + zx) + xz.

(1)

Dat P| = xy + yz + zx; P. = xz. K h i do lir ( I ) c6: P = P, + Pj

(2)

Ta

=> 1 + 2P| > ( )

x^+y^+z^^]

/g
om
.c
bo

ww

Do he

N

1

1

1

(6)

1

7
Q'j
(2)

2(x + y + z)

y-

z

xy

(3)

yz zx

Da'u bang trong (3) xay ra <=> x = y = z = 1.
'
Tir(2)(3)suy r a P >

\
xy

+

1

•+
yz zx

(4)

2(x + y + z )

Da'u bang trong (4) xay ra <::> dong thdi c6 da'u bang trong (2), 6)
X

= y = z = 1.

Do xyz = 1, nen tif (4) rut gon ta c6 P >

y=()
X =

X

O

fa

+y +z = 0

I

(x + y) + ( y + z) + (z + x )

X

+ y+ z

w.

x-+y^+z^ = l

(5)

ce

TCf (4) suy ra: m i n P. = - ^ < = > y = ()

1

D.
.
. i
1
1
1
1 1
Ro rang ta C O — + — + — > — + — + — .

(4)

2

2^ ^
2
x = -z

X

ro

(3)
..

ok

xz

(1)

y+zz+x

Da'u bang trong (2) xay ra o x = y = z = 1.

up

s/

(2)
x +y+z=0

L a i c6: P-,

x+y

P>

Ta c6: (x + y + z)' > 0 =:> x ' + y ' + z ' + 2(xy + yz + zx) > 0

x^+y^+z^ =l

1
.4

I T h e o bat dang thufc S vac-set, la c6 P >

HU('/Hg dan gidi

TiT (2) suy ra: m i n P| = - — <=>

HUi'fng ddn gidi

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p;

.

Bai 2 1 . Cho X , y, / la ba so thifc thoa man: X ' + y ' +

P,

z"*(z + x )

,, ..

X , y cung dau

1

h •

y"*(y + z)

x| = 2 ; y | = l .

Vay max P = 7 <=>

1

+ —•

2

i t

L a i theo ba't dang thilTc Cosi, ihi x + y + z > 3^/xyz = 3.

(5)
(6)

Dau bang trong (6) xay ra o x = y = z = 1.
-z

C O nghicm (chiing han x = — ; y = 0; z = — ^ la nghicm cua he (6))

Tir(5)(6)c6 P > ^
Da'u bang trong (7) xay ra o dong ihc^Ji c6 da'u bang trong (5) (6)

Vay theo tinh chat 5 cua gia t n \(1n nha't va nho nha't ( X c m bai 3 - chtfc/nj; 1

O X =y =z=1

cuon sach nay), la c6:

Vay minP = ^ < = > x = y = z = l .

min P = min P| + min P: = - - - - = - 1 .
2 2

(7)