Chuyên đề bồi dưỡng HSG GTNN GTLN(Thầy phan huy khải)
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P.GS - IS PHAN HUY KHiH
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Chuqen de
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Gia tri nli6 nlid
^Danh clio hoc sinh Idp
>BfensoantheonOidun^va
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Ta
BOI DUONG HOC SINH
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c^utrucd^tliicuaBOGDfiflT
• ^ V I E N Tl'NHBiNHT
OK]
Ha NQI
NHA XUAT BAN DAi HQC QUOC GiA HA NQI
IJCU N6I
N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI
16 Hang Chuoi - Hai Ba Trang - Ha Npi
Dien t h o a i : Bien t a p - Che ban: (04) 39714896
Hanh chinh: (04) 39714899; Tong bien tap: (04) 39714897
Fax: (04)39714899
i>Au
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Ta
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Bai todn tim gid tri U'fn nhd't, nhd nhd't ciia ham so noi rieng vd hat dang thiic ndi
chung Id mot trong nhifng chii de quan trong vd hu'p dSn tnmg chutfng trinh gidng day vd
hoc tap In) mon Todn d nhd trudng phd thong. Trong cdc de thi mon Todn ciia cdc ki thi
vdo Dai hoc, Cao dang 10 nam gun day (2002 - 2011) cdc hdi todn lien quan den vi^c
tim gid tri
nhd't, nhd nhd't ciia hdm .w thudng xuyen cd mgt vd thut'fng Id mot trong
nhiing cdu kho nhd't ciia de thi. , , ., ,
Chiu trdch nhiem xuat ban
Vdi li do do cdc cud'n sdch chuyen khdo ve chii de nay ludn luon thu hut su chii y vd
Gidm doc • Tong bien tap : T S . P H A M T H I T R A M I
quan tdm ciia ban doc. Tnmg cud'n sdch "Cdc phUtfng phdp gidi todn gid tr\ nhd't,
gid tri nho nhd't" nay, chiing toi se cung cap cho ban doc nhvtng cdch gidi thong dung
nhd't doi vdi nhiing hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so.cdng nhu hiet
Bien tap vd sita bdi:
H A I NHtf
cdch
dp dung hdi todn nay de gidi nhieu hdi todn lien quan den no.
Che ban:
Cong ty K H A N G V I E T
Noi dung ciia cud'n .sdch dUOc trinh hdy trong chUcfng.
Trinh bay bia :
C o n g ty K H A N G V I E T
Chiiong 1 v(H tieu de " Vdi bdi todn md ddu ve gid tri l^n nhd't va nhd nhd't cua ham
so"
se gidi thi^u vdi ban doc bdi todn tim gid tri Idn nhd't, nhd nhd't ciia hdm .sd'thong
Chiu trdch nhi^m ngi dung vd ban quyen
qua vi^c trinh hdy tinh da dang ciia cdc phUcfng phdp gidi hdi todn ndy. Bdng cdch diem
Cong ty TNHH MTV DjCH Vy VAN HOA KHANG VI^T
lai nhiing .sU cd m$t ciia cdc hdi thi ve chii de ndy cd mdt trong cdc ki thi tuyen .nnh Dai
hoc
- Cao dang cdc ndm tic 2002 den 2011, cdc ban se thd'y duac sU can thie't cua vi$c
Tong phdt
hdnh:
phdi trang hi cho minh nhvtng kien thiic de gidi quyet cdc hdi todn d'y. Cud'i chUtfng 1 Id
cit sd li thuyet ciia hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so. Phun nay giup
cdc ban nhiing kien thiic chud'n hi can hiet di' doc tiep cdc chUifng sau ciia cud'n sdch.
C6NG T Y TNHH MTV
Cdc phUcfng phdp ca ban vd thong dung nhd't de gidi bdi todn tim gid tri Idn nhd't vd
Sm ajP
D ! C H vy V A N H 6 A K H A N G V I | T
nhd nhdt ciia hdm sd'duac trinh hdy tit chUOng 2 den chuang 6.
., •
/^Dia chJ: 71 Dinh T i § n Hoang - P D a Kao - Q.1 - TP.HCM
~ ^
Chitang 2: Phi/mg phdp h&t ding thuCc tim gid tri l^n nhdt vd nho nhdt cua ham sd.
Dien thoai: 08. 39115694 - 39105797 - 39111969 - 39111968
ChiiOng 3: Phiicfng phdp liifng gidc hoa tim gid tri l^n nhdt vd nho nhdt cua hdm
Fax: 08. 3911 0880
Email: l
Website: www.nhasachkhangvlet.vn
^
Chitang 4: PhiiOng phdp chieu bien thien hdm sd tim gid tri Idn nhdt vd nhd nhdt
cua hdm sd.
SACH LIEN KET
ChiMng 5: Phiicfng phdp mien gid tri hdm sd tim gid tri Idn nhdt vd nhd nhdt cua
CHUYEN DE BOI D J O N G HQC SINH G 1 6 1 GIA TRI LdN NHAT, hdmsd.
ChUOng 6: PhUmg phdp dS thi vd hinh hgc tim gid tri Idn nhdt vd nhd nhdt cua
GIA TRj NHO NHAT.
hdm sd,
Ma so : 1 L-31 7DH2012.
d mSi chuang, chung toi cdgdng truyen tai den ban doc n^i dung co ban cua phuc/ng
So lugfng in 2000 Wn, kho 16x24 cm.
phdp, dUa ra cdc Idp hdi todn md phuc/ng phdp gidi no la thich h(fp nhd't. Thdng qua vifc
In tai Cty TNHH MTV in an MAI THjNH DL/C.
Phdn tich, hinh luqn vd dUa ra lam doi chiing nhieu phUtfng phdp khdc nhau gidi cUng
Dja chl: 71 Kha Van Can, P.Hiep Binh Chanh, Q.Thu Dufc, Tp.HCM.
mQt bdi todn se giup cdc ban tim duoc cho minh mQt phuang phdp m vi$t nhdt de gidi
So xuat bin: 1297-2012/CXB/08-213/DHQGHN, ngay 26 thang 10 nam 2012 hdi todn gdp phdi. Do Id dieu mdi me cua cud'n .sdch ndy. Chung toi ludn ludn gia tinh
Quyet djnh xuat b i n so: 311 LK-TN/QD-NXBDHQGHN
thdn chii dao d'y trong tvCng phdn ciia cud'n .sdch.
in xong va n6p liAi chieu qui I nam 2013.
Cty TWHH MTV D W H Khang Vi^
ChMng
7 danh de trinh hay vi$c ling dung ciia hai todn tint gid tri U'fn nhd't, nhd nhat
MdDltUVfGliHllllhllllllt
trong vi^c hi$n ludn phu
VANHiNHltCUAHAnSdr
rdng day cdng Id mot chii de thi/dtng xuyen xud't hi^n trong cdc de thi tuyen sink vao Dai
hoc - Cao dang nhQng nam gdn day (2002 - 2011).
Phdn ddu ciia chiMng 8 vc'fi tieu de "M$t sobai todn khdc tint gid tri
"hat vd nhd
§ 1 . VAIBAITOANMdDAU
nhd't cua ham so" de cyp den hai todn tim gid trf Idn nhat vd nhd nhd't ciia ho ham so
Cudn sdch nay chu yeu trinh hay cdc hai todn tim gid tri
.so
T r o n g m u c n a y c h u n g toi gidi thieu v a i bai toan v e gia tri Idn nha't va nho nha't
tham so.
ciia h a m so. T h o n g qua nhffng hai toan nay, c h u n g toi muon d e c a p d e n c a c
nhat, nhd nhcft trong Dai
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phu thuQC
phtfdng p h a p c d ban nhat d e giai c a c bai l o a n v e gia trj Idn nhat v a nho nhat se
vd Gidi tich.
di/dc trmh b a y k y y o n g c u o n s a c h nay.
Bdi todn tim gid trf U'tn nhd't, nhd nhd't trong .w hoc, hinh hoc to h(tp, hinh hoc khong
B a i t o a n 1: (De thi tuyen sink Dgi hoc, Cao dang khdi B)
gian, hinh hoc phdng, luang gidc,... se duoc chung toi trinh hay trong mot cudn chuyen
khdo khdc (sdp xud't hdn). Tuy nhien trong phdn hai ciia chU(fng 8, chung toi van ddnh
C h o h a m so' y = x + V 4 - x ^ . T i m g i a trj Idn nhat v a nho nhat c u a h a m so'
mot it trang de diem qua mot .id thi du tieu hieu ddc sdc ciia cdc hai todn nay.
n a y tren m i e n x a c djnh c u a no.
Chung toi thiet nght cudn sdch nay .se ddp dng dUilc mot sd lUOng Win hqn doc. Cdc
Hildiig dan giai
ban hoc sink phd thong, cdc thdy to gido day Todn deu cd the tim dU(/c cho minh nhCtng
Cflc/i 7 ; (PhU'dng p h a p bat d a n g thtfc)
dieu hd ich khi doc no.
Ta
s/
khdi lU(/ng U'fn cdn truyen tdi, cudn .sdch khdng the trdnh khdi cdc khiem khuye't.
up
Tdc gid rat vui long neu nhdn ditifc su gdp y ciia hgn doc, nhd't la cdc hgn ddng
nghi$p xa gdn de quyen .sdch tdt hifn nQa trong cdc idn tdi hdn tiep theo (vi chiing toi
x > - 2 ; V 4 - x ^ >0 V x
D o do f ( x ) > - 2 , V x e
ro
f(-2) =-2
G
j>
[-2;2] ,
[-2; 2]
'^'fn -
;
(1)
; i,
H;
:J
nm)
M
(2)
T i l f ( l ) ( 2 ) s u y r a m i n f ( x ) = -2.
om
T a se chu-ng m i n h f(x) < 2V2
.c
PHANHUYKHAI,
V i p n T o a n hoc, 18 DiTcfng Hoang Quoc V i ^ t - Quan C a u G i a y - H a Noi.
That vay (3) o
ok
X i n chan thanh c a m dn.
V x e [-2; 2]
X + V 4 - x ^ < lyfl
o
(3)
V4-x^ < 2 > ^ - x ' — -
c ^ 4 - x ^ < (2V2-x)^ ( d o x < 2 ) o 2 x ' - 4>/2 x + 4 > 0
bo
Tacgia
ce
«
(X -
'
72 )^ > 0.
w.
fa
Tur ( 4 ) suy ra ( 3 ) dung. Nhu" v a y ta c 6 f(x) < 2^/2
L a i cd
ww
-
Tacd
L a i c6
/g
nght rdng chdc chdn cudn sdch nay ton duc/c tdi hdn nhieu idn).
!
s
'
H a m so' d a c h o x a c d i n h k h i -2 < x < 2.
Mat ddu vc'fi tinh than nghiem tiic, ddy trdch nhi(m khi viet cudn sdch nhung vdi mot
Thtf tCf gop y xin guTi ve theo dia chi sau:
'
Nhdn
f(V2) = 2N/2 , n e n
max f(x) =
-2
xet:
(4)
V x e [-2;
2].
2V2 .
,
1. C a c h g i a i tren h o a n toan dtfa v ^ o ba't d a n g thtfc, n e n ngiTdi ta thiTdng g o i la
I,'
,
i
phi/cfng phap bat d a n g thiJc.
2. T a c d the sOr d u n g ba't dang thifc B u n h i a c o p s k i d e g i a i nhu" s a u :
1 ;'
T h e o bat d a n g thi?c B u n h i a c o p s k i ta c d :
x.l + V 4 - x ^ l l
<[x^+(4-x^)|(l^+1^)
= > x + V 4 - x ^ <2sf2.
(5)
Chuyen
BDHSG Toan gia tr| Idn nha't
Cty TNHH MTV DWH Khang Vi§t
g'A tr| nh6 nhat - Phan Huy KhJi
+ 1 + 1 > 3z.
Tir do va diTa vao gia thie't x + y + z = 3 suy ra:
x-y/x^ + 8 y z
D a u bang trong (2) xay ra o X = y = z = 1.
(3)
D a u bang trong (3) xay ra <=> dong thdi c6 dau bang trong (1), (2)
<=>X = y = Z = l .
x^jy}
Z^.,-,
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y^
y +yz + z
2
2
x +xy + y
2
Hiidngddngiai
;;':..(•'•...!;
'-„4/.!'
j,,
= (x + y + z ) ^ x " ' + y ^ + z ' ' + 2 4 x y z j .
4
I
+
y^^z^+zx + x ^ j
4
.
*
up
om
(2)
Theo bat d i n g thtfc Cosi, ta c6:
ok
.c
^
^
. .
y + y z + z X j + ( x y ) ( y z ) + (xy)(zx) + (yz)(zx)
i
ce
x' + y' + z ' > x y + y V + z V
(4)
fa
x^y^ + y^z^ + z^x^ > (xy)(yz) + (xy)(zx) + (yz)(zx)
(3)
3(x^y^+y^z^+z^x^]
—
— ( hay P > 1.
3(x^y^+yV+z\^j
4^-
ww
Tir (2), (3), (4) suy ra: P >
w.
z" + z'' + z > 3z^
(5)
De thay dau b^ng trong (5) xay ra o X = y = z > 0.
^ Vay min P = l < i > x = y = z > 0 .
>/y^ + 8zx
+ ^xy
Hiidng ddn giai
V i e t l a i P difdi dang sau:
(3)
,, .
(4)
• ,
, .
Nhqn xet: Ta c6 bai toan IMng
I
,
^^ ^
= 1.
(x + y + z)^
(5)
Is!,. Jr. ;
De thay da'u bang trong (5) xay r a o x = y = z = 1.
tU" sau:
Cho X > 0, y > 0, z > 0 va X + y + z = 1. T i m gia tri nho nhat cua bieu thtfc
P =
x''+8yz
-+ •
y-^+8zx
z^+8xy
Ta giai nhu" sau: P = —
+
X' + 8 x y z
y +8xyz
A p dung ba't dang thuTc Svac-xd, ta c6: P >
z^
/: + 8 x y z
(x + y + z)
x' + y + z" + 24xyz
(*)
(**)
Theo bai tren ta c6: (x + y + z ) ' > x V y ' + z ' + 24xyz.
Tilf (**), (***) suy ra: P >
Bai 15. Cho x, y, z la cac so thifc diftftig. T i m gia t r i nho nhat ciaa bieu thuTc
+8yz
+ z' + 24xyz.
hay (X + y + z ) ' > x ' +
Thay (3), (4) vao (2) va c6: P >
ro
x^+y^+z^+2(xy+yV+zV)
2^x
•
'
> x ' + y-'+ z ' + 277xyz.\/xVz^ - 3xyz
bo
hayP>
'
(X + y + z)^ = x^ + y ' + z' + 3(x + y + /)(xy + y/ + /x) - 3xyz
z^(x^ + xy + y ^ j
A p dung bat dang thufc Svac-xd, ta c6:
fx^+y^+z^f
P>
i
L
x^(y^ + y z + z^) + y^ (z^ + z x + x ^ j + z^(x^ + x y + y^)
,
(2)
-.
*
A p dung ba't dang thiJc Cosi, ta c6:
.
(1)
Ta
x ^ ^ y ^ + y z + z^j
y_
s/
+
/g
4
^1
.
(x + y + z) x ( x ' + 8 y z ) + y ( y ^ + 8 z x ) + z ( z ^ + 8 x y )
V i e t l a i P dtfdi dang:
P=
,
X/X.N/X^X^ + 8 y z + ^/y.^/y^/y^ +8zx + N/Z.VZI/Z^ + 8 x y
z^
z +ZX + X
+ 8yz + y ^ y ^ + 8zx + z-y/z^ + 8xy
A p dung bat dang thiJc Bunhiacopski, ta c6:
B a i 14, Cho x, y, z la cac so' thifc di/dng. T i m gia tri nho nhat ciaa bieu thtfc
2 2
z^z^ + 8 x y
(x + y + z ) '
'
V a y m i n P = 1 <=>x = y = z = l .
2
y-y/y^ + 8 z x
TO'(1) va theo bat dang thiJc Svac-xd, ta c6:
T i i r ( l ) , ( 2 ) s u y r a P > 1.
r.
(I)
P-
(2)
+ y ' + z^ > 3.
(x + y + z)^
" \y P >
(x + y + z)-
1=
1.
x +y + z
Vay min P = 1 o X = y = z = ^ .
^ a i 16. Gia siif x, y, z la ba canh cua mot tam giac c6 chu v i bang 12.
1 ^ I
ChuySn
BDHSG Toan gia tri I6n nha't va g\& tri nh6 nha't - Phan Huy KhSi
Cty TNHH MTV DVVH Khang Vijt
Dau bkng Irong (5) xay ra <=> x = V 4 - x ^ o x = V2 .
- 2 < F((p) < 2V2 V - | < ( p < ^ ,
Vay maxy = y/l C:>x = yl2 .
Cdch 2: (PhiTOng phap chieu bie'n thien ham so)
Xet ham so f(x) = x + V4 - x^ vdi - 2 < x < 2
F((p) = - 2 <=> cos
v4-x^-x
I I S
/4-x^
Vay
\/4-x'
X
'5
< 2, ta c6 (4 - x") - x" = 4 - 2x-.
9
= — <=> X = N/2
4
V2
71
CP--
2
i
max Hx)=
max
-2
R6 rang k h i - 2 < X < 0, thi f'(x)> 0.
Xet khi 0 <
= 10
71
<=> (p — =
^ 4 4
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X
Ta CO l"'(x) = 1
71
F((p) = 2%/^ <=> cos
min
• -,
71
o
(p = — <=>x = - 2 .
'
2
F((p) = - 2 .
<(n<
2
Do 4 - 2x' > 0 khi 0 < X < yfl va 4 - 2x' < 0 khi N/2 < x < 2, nen ta c6 bang
37t
•
F((p) = 2>/2;
min
~
;
Tt
7t
f(x)=
,
2
Cdch 4: (Phu'dng phtip mien gia trj ham so)
bien thien sau:
0
^/2
Gia sur m hi mot gia trj tiiy y cua ham so \'(x) = x + \l4-x'
^^^'..2.
Khi do phu'dng trinh an x sau day x + \J4-x~
(1) c6 nghiem.
\ / 4 - x - = m - x. (2)
Ta
Ro rang (1) o
=m
"
i
min l(x) = min{l'(-2);r(2)) = min(-2;2) = - 2
/g
- 2 < ,\ 2
>
up
max l"(x) = I(N/2) = 2N/2 ;
:
ro
•"''^
s/
B^i loan ltd thanh: Tim m de (2) c6 nghiem.
Tirdosuyra
fa
w.
F((p) = 2sin(p + >/4(l - sin" (p)
2sin(p + 7 4 c o s ' cp = 2sin(p + 2|eos(p
ww
TCr do ta quy ve xet ham so
= 2sin(p + 2cos(p (do khi - - ^ < cp < ^ ihi coscp > 0)
De lha'y y = m - x o x + y = m, con y = ^ 4 - x " c^< ^
I y>0
Dc tha'y dieii nay xay ra
khi
va chi khi du'cVng
thring X + y = m nam
giiJa hai du'clng x + y = - 2
va X + y = 2 V2 , ti'fc la
khi vii chi khi - 2 < m <
2V2 .(3)
= 2N/2COS((P--).
,^
n
n
Tir(3)suyra
37t
71
rt
Do — < ( p < - => - • — < ( p - - < - .
2
2
4
4 4
Tif do suy ra - — ^ < cos
f(P--
;,i;.f-
max r(x) = 2%/2;
-2
min
<1
~^
(phan nam phia tren Iriic hoiinh cat nhau).
ce
Xet ham so t(\) = \+ ^ - x ' \(U - 2 < x < 2
Do - 2 < X < 2, nen dat x = 2sin(p vdi - - ^ < cp < ^ .
(2) CO nghiem khi vii chi khi difclng cong y = SJA-X^ va diTcJng thang y = m - x
cat nhau.
n . . . : , . v - , ! - . - x ,
Vay ta can tim m de du'c'Ing thang x + y = m va nifa du^clng Iron x^ + y" = 4
bo
Cat7i J; (PhifcJng phap UMng giiic hoa)
'
i
ok
trinh bay cJ' Iren.
^"
.c
om
Nhgn xet: Ten goi cua phifdng phap hoaii toan phan linh di'ing qua each giiii vifa
vdi - 2 < x < 2.
-2
f(x) = - 2 .
Chuygn dg BDHSG Toan gia trj Icin nha't
Cty TNHH MTV DWH Khang Viet
gia tri nh6 nhat - Phan Huy Khtii
Nhdn xet: Cach giai tren diTa vao each tim gia tri cua ham so
(x-y)(l-xy)
day c6 ket hdp
T i i r ( l ) ( 2 ) suy ra
them phifdng phap suT dung do thi va hinh hoc), vi the ta c6 the noi rang da
sur dung phi/dng phap mien gia tri ham so' de giai bai toan tim gia trj \6n nhat
(x-y)(l-xy)
bai loan tim gia trj Idn nhat va nho nhat cua ham so. M o i phU'dng phap deu
Bai toan 2: Cho x > 0, y > 0. Tim gia tri Idn nhal va nho nhat cua bieu thiJc P =
(l + x ) ^ l + y)2
•
'
Lai
•
(De thi tuyen sink Dai hoc, Cao ddn^ khoi D )
(
. i
M
I
CacA 7; (Phufdng phap ba'l dang ihiJc)
(1 + x)^
4
X>
TiTOng liTlai c6
'f'
'<
1
0, y > 0. P = -
x = 1; y = 0.
up
i5«»f^« 'flfffetv''}l>_'n^frflJ j s l i i i . f U '
X
^ (y-1)'
(1 + y)^ " 4
4
(2)
Tom lai max P = — < = > x = l ; y = 0; minP =
ww
w.
Do X > 0, nen liJT (2) suy ra P > - - V x > 0; y > 0. P = - - o x = 0; y = 1.
<=>x = ();y = 1.
(x-y)(l-xy)
(l + x ) ^ l + y)2
X
x = l ; y = 0, khi d6P = [xy = ()
x + y = l + xy
[x + y = l
x = 0;y = l , k h i d6P = - i
(1) ,
Dafu bang trong (2) xay ra o xy = 0.
4(l + x ) ^ l + y)-
_ 1
4(l + x)2(l + y)^
4
:
4y
1--
(i + y r
<4
(doy>0)
Tird6suyraP^^^-y^/^-^^[4 Vx.O;y^O
(l + x)^(l + y ) ' 4
Mat khac P = - <=>' ^ ^ . VaymaxP= - <=>x = l ; y = 0.
x=l
4
4
Do vai iro binh dang giffa x va y, nen la co
(l + y ) 2 ( l + x)'
4
. p _ , U - y K l - x y ) > _ ivx>(),y>0.
(l + x)^I + y)^
Mat khac P - - - o x = ( ) ; y = 1. V a y n e n P =
4
Cach 4: (Phifcfng phap lifting giac hoa)
1
4
o x = 0 ; y = 1.
(2). .i
'1
i
(xem each 1).
Ta co: P =
Do X > 0; y > 0, ncn hien nhien la c6
x - y | | l - x y | < ( x + y ) ( l + xy)
ta co: ( x - y ) ( l - x y ) ^ ( x - y . 1 - xy)
2/1 , . , \
n j_ v^2/-l . v"!^
(l + x)"(l
+ y ) ' A4(l
+ x ) ' ( l + y)
( x - y + l - x y ) 2 ^ (l + x ) 2 ( l - y ) '
Lai eo
• '
nhau bai loan tren.
- y 1 - xy
(l + x)^(l + y)^
(5)
Nhdn xet: Cung suT dung phiTcMg phap ba't dang thiifc, nhu-ng ta co 3 each giai khi
Cach 2: (PhiTdng phap bat dang ihu-c)
Ta c6:
hay - i < P < | .
4
4
xy = 0
(y-x)(l-yx) ^ 1
fa
(1 + x)
1
(1)
{\ yf
ce
P=
y
ro
4(l + x)2
Do y > 0, nen lij" (1) suy ra P < - , V
^'
,
/g
4 ~ 4
2
s/
(1 + y)^ ^ 4
1 - i _ (x-ir
(1 + y)
4
AB .
1
.c
4(1+ x)'
y
om
4 x - ( l + x)'
P
Ta
1
ok
(1 + y)^
X
bo
(1 + x)^
P =
_
CO
CacA J ; (Phu'dng phap ba'l dang Ihtfc)
lha'y P c6 the vie't lai dudi dang sau day
y
(4)
Tom lai maxP = - < = > x = I ; y = 0; minP = — <=> x = 0; y = 1.
4
4
HuAng ddn giai
X
(l + x ) 2 ( l + y)^
iL
ie
uO
nT
hi
Da
iH
oc
01
/
Tir(3) & (4) di den
CO nhffng ifu diem rieng cua no.
•• ' •
(3)
•
Dau bang trong (4) xay ra « x + y = 1 + xy.
Binh ludn: Vdi bai loan 1, la da su* dung bon phu'cfng phap khac nhau de giai
•: ^
+ y)(l + xy)
Mat khac d i lha'y [(x + y) + (1 + xy)]^ > 4(x + y)(l + x y ) .
va nho nhat noi tren.
(x-y)(l-xy)
(l + x ) ' ( l + y)^
(X
i2
[(x + y) + (l + xy)]
(1 + x)^
(1 + y)^
Do X > 0; y > 0, nen dat x = tan^a, y = tan^(3, ( ) < a < - ; 0 < p < - -
,
(1 + tan^ar
,
^ ^ " ' f , = lan^acosV -
(1 + lan^ p)^
- - < P < 4
4
L a i l h a y P = ^<=><
P=
—
4
«
sau d a y ( a n t )
— sin'2p.
(1)
-•^'^ "^'^^ = m
t^+2t + 3
Va, pG|();-).
2
nen (2)
n
sin 2a = 1
sin2p = ()'^
X
p =o
y = ()'
4 <=> i
- 1
*
x-O
sin2p=I
m i n P = - - ^ <z> x = 0 ; y = 1 .
i*>
s/
c u n g siir d u n g phiTdng p h a p ba'l d a n g ihiJc ( b a e a c h n a y l a i k h t i c n h a u ) . Q u a
, , „,,
ro
n h o nha't eiia h a m so.
ddn gidi
'
X e t h a i k h a n a n g sau:
.i
om
.
< {i
N e u m^l,
X
- 6 < m < 3.
V i i y ( 3 ) CO n g h i c m k h i v a e h i k h i - 6 < m < 3.
,/ /
D o m l a g i a t r j t u y y c i i a r(t), n e n t i i " ( 4 ) suy r a
,r
Ket hdp
, 1 !
^
Cat7i 2 ; (PhU'dng p h a p m i e n g i i i t r i h a m so)
'
D o X ' + y " = 1, n e n l a d a l x = s i n a . y = c o s a , v d i a G |(); 271].
, .
KhidoP=
2sin" a + 12sintteosa
1-cos2a+ 6sin2a
— =
^ 1 + 2sinacosa +2cos'a
sm2a + cos2a + 2 .
G o i m l a g i a t r i t u y y c i i a P.
K h i d o phu'dng t r i n h sau d a y ( a n a )
—
/ \
-2V3
t"+2t +3
, (1
X
1-cos2a + 6sin2a
—=m
sin 2 a + cos 2 a + 2
;
day I = — va t
y
e
(l)
Cdch
(3)
( 6 - m ) s i n 2 a - (1 + m ) c o s 2 a = 2 m - 1.
m ) " + (1 + m ) ' > ( 2 m - 1)"
3: (PhiTctng p h a p c h i e u b i e n ihiC-n h a m s o )
TacoP^
2(x^.6xy)^
X
*
,
1 - cos2a + 6sin2a = m(sin2a + cos2a + 2)
-> •.,
(4)
T i r d o suy ra m a x P = 3, m i n P = - 6 k h i x - + y~ = 1.
.
,
, ,
c ^ 2 m - - 3 m - 9 < ( ) o - 6 < m < 3 .
;
..^
C O n g h i t M i i . D o |sin2a + c o s 2 a | < V2 , V a e |(), 2TC|
o
,
'
V d i d i e u k i e n x ' + y^ = 1 i h i m a x P = 3, m i n P = - 6 .
Tir do (2)o
+ 6
(4)
P = 2 k h i y = 0, l a d i d e n k c l l u a n :
(3) CO n g h i e m 0 ( 6 -
2t- + i2t
X
k h i d o (3) c 6 n g h i e m k h i va c h i k h i A ' > 0
=> s i n 2 a + c o s 2 a + 2 > 0 V a 6 |(); 27i|.
(1)
1. N e u y = 0 ( k h i d o x = 1). L u c n a y P = 2.
2. N c u y ^ 0. K h i d o P =
N e u m = 2 , k h i d o 2 ( m - 6 ) ^ 0, nen ( 3 ) c 6 n g h i c m . V a y m = 2 la m o t g i a
ww
x^ + 2 x y + 3y^ '
fa
•'
w.
2(x^+6xy)
+ y " = 1, n e n ta e o : P =
ce
1: ( P h i f d n g p h a p m i e n g i a t r i h a m so)
X'
.c
ok
(De thi tuyen sink Dai hoc. Coo ddn)> khoi B)
Hitdng
Do
+6xy)^
1 + 2xy + 2y"
bo
,
/g
IJai t o a n 3 : G i a silf X, y la hai so ihifc sao c h o X " + y^ = 1.
':dch
up
d o la t h a y r o l i n h d a d a n g c u a phiTdng p h a p d u n g d e t i m g i a t r i lofn nha't v a
' ('
(3)
:
m a x P = max r(t) = 3 v a m i n P = m i n r(t) = - 6 .
y*()
ItR
y*l)
lelR
V d i biii loan t r c n la c 6 4 each g i a i khac nhaii, Irong d o c 6 3 each
T i m g i a t r i UKn nha't v a n h o nha't c i i a b i c u thiJc P =
• » 2 1 ' + 12l = m ( t ' + 2t + 3 )
c:> m ' + 3 m - 18 < 0 o
y = i
V a y m a x P = ^ < z > x = l ; y = ();
( 2 ) C O n g h i c m . D e tha'y v i t ' + 2 l + 3 > 0 V l ,
t r i c u a h a m so r(t).
*
a =0
sin 2 a = 0
:'
<=>(m - 2 ) t ' + 2 ( m - 6)1 + 3 m = 0 .
a =—
k h i d o phiTcIng t r i n h
t^+2l4-3
.
= sinWos^a-sin'Pcos'P=-sin'2a-
TOr(l)suyra
luan:
•
-li—Ili^,
G o i m la g i a t r i l u y y c i i a h a m so 1(1) =
tan^pcos^p
iL
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K h i do P =
liinh
Cty TNHH MTV DWH Khang Vi^t
BDHSG To^n gi^ tr| Idn nha't vA gia trj nh6 nha't - Phan Huy Khii
Ta
ChuySn
j
(xemcachl).
+ 2xy + 3y"
N e u y = 0, l h i P = 2.
"
;
Cty TNHH MTV D W H Khang Vigt
Chuy§n dg BDHSG Tpan gi^ tr| Ifln nhat
* Ncu y ^ 0, thi P =
gia tri nh6 nhat - Phan Huy KhSi
^^^^ vdi t = y
Ap dvng (2) vdi a = ^ ; b = f.
6
Kb. do a > 0, b > 0 va ab = - . 1 (do x . y),
r +21 + 3
Dat f(t) =
t G R thi f'(t) t^+2l + 3
-81^ + 121 + 36
,
( l ^ + 2 t + 3)^
, 2t^-3t-9
4
( t ^ + 2 t + 3)^ '
1+^
V
Ta CO bang bie'n thien sau:
2
-00
ii
iH:
3
l ^ ' ^ 1+
z
z _
+O0
iL
ie
uO
nT
hi
Da
iH
oc
01
/
I
(4)
I_+-L->-^
nen ta c6:
Dau bang trong (4) xay ra
0
I'd)
1(0
+
-
0
2
Tif (4) ta
Tir do suy ra max f(l) = 3 va min t'(t) - - 6 .
teE
'f.f
U'W 'n'.l-.U:^:''
Vay maxP = 3, minP = - 6 khi x^ + y ' = 1.
Ta
va nho nhat cung the hicn ro qua thi du nay.
up
2x + 3y
z
z+X
ro
thiJc: P =
s/
Bai toan 4: Cho x > y, x > z va x, y, z e [ 1 ; 4].Tim gia tri nho nhat ciia bieu
/g
(De thi tuyen sink Dai hoc Cao ddiifi khoi A - 2011)
om
Hii(fng dan gidi
.c
LtJi giiii cua bai toan nay la su" kel hctp khco Ico cua hai phifcfng phap bat
1
2 + 3^
' ' '
'
•
X
+
1+^
y
(1)
Xet ham so' f(t) =
Taco: f'(t) = ^
1 + '^
ww
Tru"(1c hel ta chiJng minh bat dang thiJc sau: *
" ' *'
1
1
2
Neu a > 0, b > 0 va ab > 1, Ihi la c6:
+
>
1 + a 1 + b 1 + x/ab
VI t > 1 ^
I
(2)
f'(t)
f(t)
Dau bang trong (2) xay ra khi va chi khi a = b hoac ab = 1.
Vay
Thatvay(2)
> 0 o
1+a
l + >/abJ
1^1 + b
1 + Vab
Tab
(l + aKl + x/ab)
(V^-^/b)^^/^-l)
>0.
(l + a)(l + b)(l + >/ab)
Do a > 0, b > 0, ab > 1, vay (3) dung suy ra (2) dung.
>yab-b
(l + b)(i +
(3)
7^) >
7y^ = z
x =y
A
i
....
i
;l
I
^
^
-
+ - 1 ^ vdi 1 < t < 2.
1+t
2
(31 -61^) + ( 3 1 ^ - 4 1 ^ - 9
^
=
(21^+3)^(1 + 1)^
^
f (t) < 0 V t e [ 1 ; 2 ] . TO do c6 bang bien thien sau:
1
1
i
•
min f (t) = f (2) = | 1 . TO do suy ra P > ^ ,
33
l
<=>
Vy
\
t
2
?>—!— +—-hay P > - ^
+ 7~T3
1+t
21^+3 l + l
2+
w.
•
1
•+
bo
1
P diTdi dang: P =
ce
Ihtfc
1+
21+3
61
fa
Viet lai bieu
(5)
X
Datt = E . D o x > y v a x . y e l l ; 4 ] n e n s u y r a l < ^ < 4 = ^ l < t < 2 . K h i d 6
ok
dang ihiJc va chieu bien thien ham so nhif sau:
x=y
+
Dau bang trong (5)xayra
Binh luqn: Tinh da dang cua cac phU'dng phap giai bai loan lim gia tri Idn nhat
y
y+z
P>
X
• '
=z
2 y y
2+3^
leM
x
CO
^
y
1
2
X
^
—•—
Chuygn dg BDHSG Toan
gii
trj I6n nha't va glA tr| nh6 nha't - Phan Huy
KhAi
- Phiftlng phap mien gia tri ham so.
'i--^ j^ifif'^f^'M^^'•''•>::^r^'
- Phi/dng phap lU'dng giac hoa.
..c.^i '.-rl- Phi/ctng phap hinh hoc hoa. ,;:^^^iry,
h^> rti
- .
Cac ban cung da tha'y dtfdc chiing ta c6 the c6 nhieu phU'dng phap khac
nhau de giai cung mot bai toan tim gia tri Idn nhat va nho nhat cua ham
Do X, y, z e [ 1; 4] ncn P = — <=>x = 4, y = l , z = 2.
33
34
Nhu" the minP = <=>x = 4;y = l;z = 2.
33 ;
~"
Bai toan 5: Cho bon so ihifc a, b, c, d thoa man dieu kien a^ + b" = c' + d"^ = 5.
Tim gia trj Idn nha'l cua bieu thiJc
P = > y 5 - a - 2 b + V 5 - c - 2 d +N/5 - a c - bd .
Hii(fng dan gidi
Ldi giai hay nhat va dac sSc nhat cho bai loan nay la phu^dng phap su* dung
hinh hoc sau day:
Ta thay cac diem M(a; b), N(c; d) va Q( 1; 2) trong do a, b, c, d la cac so thifc
thoa man dieu kien dau bai deu nam tren difdng Iron c6 tam tai go'c toa do
va ban kinh bhng v 5 . '
f . i"i: V
^^
Viet lai bieu iMc P dxidi dang sau:
f ? x^vlfi^m/(a-l)2+(b-2)^
ka-cf+ih-df
P=
iL
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Da
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oc
01
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SINH V A O D A I H Q C , C A O D A N G
v2
2
=
-N/30.
. Qua 5 bai toan tren, chung toi da gidi thieu vdi c&c ban cac phiTdng phap?
chinh de giai bai toan tim gia tri Idn nhat va nho nhat cua ham so".
- PhiTdng phap bat d^ng thtfc.
- PhiTcfng phap chieu bie'n Ihien ham so.
Cac bai toan tim gia trj Idn nhat va nho nha't cua ham so thu'dng xuyen xua't
hien trong cac ki thi tuyen sinh vao Dai hoc, Cao dang nhiTng nam gan day.
Trong muc nay chung toi xin gidi thieu cac bai toan ay kem theo nhffng binh
luan can thiet.
, ,/ • ^
v
Bai 1: (De thi tuyen sinh Dai hoc Cao dunf- khoi A-2011)
Cho X , y, z la cac so thiTc sao cho x > y, x > z va x, y, z e [1; 4). Tim gia trj
^
t
X
y
z
nho nha't cua bieu thuTc: P =
+
+
.
7 "• :: -
Ta
s/
/g
om
.c
ok
bo
ce
fa
w.
P < ^ ^
ww
3^30
ro
d day CMNQ la chu vi cua tam giac MNQ.
Ta sur dung ke't qua quen bict trong hinh
hoc phiing sau day: Trong cac tam giac
npi tiep trong mot di/dng tron ban kinh
R cho trU'dc, thi tam giac deu la tam
giac CO chu vi Idn nhat. Mat khac tam
giac deu noi tiep trong du'dng tron c6
ban kinh R c6 canh b^ng R ^/3 .
Do d6 maxP =
N H O N H A T C U A H A M S O T R O N G C A C KJ THI T U Y E N
up
(MQ + NQ + MN) =
VivayCMNQ< 3 N / l 5 . T i r ( l ) s u y r a
§2. N H I N LAI C A C BAI T O A N V E GIA TRj L 6 N N H A T V A
2x
+ 3y
y+
z
z + x
• l;
' 1^)'?'
HUdng ddn gidi
Xem Idi giai trong bai toan 4, muc §1, chu'dng 1 cuon sach nay.
Binh luan:
1. Mau chot de giai bai 1 la d cho bang each su" dung mot ba't dang thiJc
phu, de diTa ve danh gia P > — ^ — + — y .
(1)
De'n day b^ng each difa vao an phu t = ^ -• vdi t G 11; 2] ta quy v^ danh
y
gii P >
+ _ L . f(t).
'^H*>v^^-;
•
2t2+3 1 + t
R6 rang tiep theo ta nghT ngay den se sur dung phiTdng phap chieu bien
thien ham so de tim min f(t) vdi 1 < t < 2.
Tir do ke't hdp hai qua trinh tren ta se di den IcJi giai cho bai toan. Van d6
la d cho viec phat hien ra (1) khong phai la dieu de dang.
Chuy6n dg BDHSG Toan gii trj I6n nha't va gi^ tri nh6 nha't - Phan Huy Kh5i
Cty TNHH MTV DWH Khang Vigt
2. Thay cho vice suT dung mot bat dang thtfe phu, ta co each lam sau day c6
ve "tiT nhien " hcJn mot chiit.
Coi P nhiT la mot ham so'eua z, xet ham so'an z.
' P = P(z)= —
+ - ^ + - ^ vdiz G [l;x].
'>;<
2x + 3 y y + z z + x
Khido P'(z) = ()
L +
(y + z)^ {z +X xf _ x(y+
{y +z)^-y(x
zfiz + +xfz)^ _
Tim gia tri nho nha't cua bieu thtfc P = 4
(1),
* N e u x ^ y , lhiP(z)= ^ + ^ ^ + - ^ = ^- V z e [ l ; x ]
5y y + z z + y 5
- .
.. i
* Ne'u X > y (chii y la x > y, nen khi x y thi x > y) thi x - y > 0 nen
ii^.'"' P'(z) = 0<=>z^-xy = 0<=>z= ^xy •
aiaj',;;:.;J.J:-:^'^
Ta CO bang bien thien sau (suy ra tif (1))
? , ! ' !7
z
1
2
P'(z)
0
+
P(z)
^ - ^ ^
— ^
.Hi!K
1
i
Da'u bang trong (2) xay ra o z = ^xy .
s/
up
ro
om
ok
fa
ce
bo
^
slx+yfy
(2)
w.
p ( z ) > ^ - ^ ^ Jy
2x + 3y
+
P = P(z)> — -1— + —2
.c
X
y
2x + 3y " y + ^ " 7^ + x
ww
p(7^)=
/g
Vay vdi mpi z e [1; x], ta c6: P(z) >
Hiidng dan giai
a b\
f a h\
a b^a b
—
— -3
Difa P ve dang sau: P = 4 Vb— + &)
+ 9 b + a— - 2
h—
b a
b a\
a b^
a b
a b
= 4 — + — + 9 b ^ a j - 1 2 —b + —a 18. (1) •
.b a j
Viet lai gia thie't diTdi dang sau: 2 r a b ' +1 = (a + b) 1 + b aJ
ab
2 '2,\/^
Theo baft dang thtfc Cosi, ta c6 I + — > - p = .
ab vab
^a b " + l > ^ ( a + b) = 2V2
—
Thay (3) vao (2) va c6: 2 vu + ay
Vb ^ Va .(4)
Vab
b a
Ta
1
r —
b^^
+ 9 b^ a^
+ )
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(x-y)(z^ - x y )
{y + zf(z + xf
Bai 2: (De thi tuyen sink Dai hoc Cao ddrtfi khoi 8-2011)
Cho a, b la hai so thiTc dUdng thoa man dieu kien:
2(a^ + b^) + ab = (a + b)(ab + 2)
6 34
Den day ta lie'p tuc giai nhiT phan sau cua bai 4, muc §1 v6i lifu y rkng do - >—
5 33
34
nen minP = —
33
Ro rang viec phat hien ra (2) theo cdch giai nay "tif nhien" hdn trong
each giai cua bai 4, mat du no phuTc tap hcfn ve mat tinh toan!
Khi do tir(4) ta c6 2(t^ - 2) + 1 > 2 > ^ t hay 2t' - 2 V2 t - 3 > 0
=>(>/2t+l)(N/2t-3)>0.
Do t > 2 =>
t + 1 > 0, nen tiif (5) suy ra
'
(5)
V2t-3>0=>t>^=>i + ^=t^-2>l.
(6)
^/2 b a
2
Bai toan quy ve:
•>^'-*v-'.^*'.'^^
5
Tim gia tri nho nha't cua ham so f(t) = 4t^ + 9t^ - 12t - 18 vdi t > - .
Ta c6: f'(t) = 12t^ + 18t - 1 2 = 6(2t^ + 3t - 2) va c6 bang xet dau sau:
1
-2
t
2
2
f'(t)
0
0
+
+
+
f(t)
23
23
(7)
Vay minf(t) = f 2. = - —4 . t t f c m t a c d P > - —
4
'4
1
i
15
Chuyfin dg BDHSG Join g'lA tr| Ifln nha't vt g\i trj nh6 nha't - Phan Huy KhAi
Cty TNHH MTV DVVH Khang Vi^t
Da'u bkng irong (7) x a y ra khi va chi khi
(diiu bang xay ra ehi lai t = 0), n c n r ' ( t ) la ham nghich bien trcn |(); - J.
b
_a _
b
• •
,
J_
Tcrdc)
V t G [0; - 1 .
r(t)>r
,b ~ 2
a
Do f
2(a^ +b-^) + ab = (a + b)(ab+ !)<=> 2(a' + b - ) + ab = (a + b ) ( a b + l )
a > 0; b > 0
"
'
^ 11 _ 2V3 >():=> r'(t) > 0 V t e [ 0 ; - |.
iL
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Da
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a = 2;b = l
a > 0; b > 0
3
nen i d ) la hiim dong bien trcn t e |{); - ] .
" '
a = l;b = 2
. !.
^ ,
i ,, ,
Tir do suy ra 1(1) > 1(0) = 2 V t e (0; ^ ].
23
V a y minP = — ^ khi va ch'i khi a = 2, b = 1 hoac a = 1, b = 2.
Nhif the ta eo M > 2 V I e |0; ^ \
Binh
ludn:
M =2khi va ehi khi ab = be = ca; ab + be + ea = 0; a + b + c = 1 tuTe la khi va
ehi khi (a; b; e) la mot trong eac bo so ( 1 ; 0; 0), (0; 1; 0) va (0; 0; 1). D o do
V i c e di/a P ve dang ( I ) la Ic liT n h i c n . Cai kho la t i m m i e n xac dinh cua bien
gia tri nho nha't ei'ia M la 2.
Binh ludn: V i c e x c t dai lu'dng phu thu()e vao bien ab + be + ea la mot y nghia
a
Ta
b
hoan toan tiT n h i c n . D i c u do dan den vice x c t eac he thuTc (1) va (2). D e n
ro
so" ham so dc giai bai loan.
Va^Tb^TiJ .
om
M = 3(a'b^ + b ' e ' + e'a^) + 3(ab + be + ca) + 2
.c
/
/g
B a i 3: Cho eac so' thi/e khong am a, b, c thoa man a + b + e = 1.
T i m gia I n nho nha't cua bicu ihtfe:
ok
(De thi tuyen sinh Dili Iwc, Cat) dcinii; khoi B ~ 20J0)
3 ( a ' b ' + b'c^ + c'a') > (ab + be + ca)".
"
ce
.t
fa
Ro rang ta CO bat ddng thiJe hien nhicn sau
(1)
w.
ww
va suf dung phiTdng phap chieu bien thicn ham so de giai b a i toan la mot
vice lam tiT n h i c n ! Cach giai vifa trinh bay ro rang la each giai l o i U\ nha't.
B a i 4: (De thi tuyen sink Dai hoc, Cao clan,^ khoi D - 2010)
T i m gia t r i nho nha't cua ham .so y =
+ 4 x + 21 - V - x ^ + 3 x + 10 trcn
HuAng ddn giai
Ta l a i eo a^ + b ' + e' = (a + b + e ) ' - 2(ab + be + ea) = 1 - 2(ab + be + ca). (2)
Tir(l)(2)suyra
M > (ab + be + ea)^ + 3(ab + be + ca) + 2 ^ 1 - 2 ( a b + be + e a ) .
1
day vice xet ham so': r ( t ) = t " + 3t + 2 V l - 2 t vdti 0 < t < -
mien xac dinh eija no.
bo
HUcing ddn giai
up
d i c u k i c n (6). Con l a i d l nhicn la silr dung phu'dng phap ehicu bien t h i c n ham
s/
Bang each kc'l hdp khco leo giffa dieii kien va bat dang thiife Cosi la suy ra
(3)
H a m so' xac dinh k h i thoa mfin he sau:
-x^+4x +21>0
-x^ +3x + l ( ) > 0
<=>S
' "
'»
f-3
<:> - 2 < X < 5
- 2 < X < .S
(1)
Da'u bang Irong (3) xay ra k h i va chi khi c6 dau bang trong (1), ttJc la k h i va
V i ( - x ' + 4x + 2 1 ) - (-X- + 3X+ l()) = x + 11 > 0 , (suy t u r ( l )
chi khi ab = be = ca.
vay y > 0 V - 2 < X < 5,
.
,
Dat t = ab + be + ea. Ta eo 0 < t < -(a + b + c)^ _ 1
3
~ 3 "
TCr (3) suy ra x c t ham so sau: f ( t ) = t^ + 3t + 2 N / 1 - 2 I
Ta CO y- = (-x^ + 4x + 2 1 ) + (-X' + 3x + 10) - 2 7(-x^ + 4 x + 21)(x^ + 3x + 10)
vc'Ji 0 < t < ^
= (X + 3)(7 - X) + (X + 2)(5 - X) - 2 V(x + 3 ) ( 7 - x ) ( x + 2 ) ( 5 - x ' )
= (x + 3 ) ( 5 - x ) + (x + 2 ) ( 7 - x) + 2 - V(x + 3 ) ( 7 - x ) ( x + 2 ) ( 5 - x )
T a c o : f ' ( l ) = 2t + 3 +
Vl-2t
r(t) = 2 -
(2)
<()
N/(l-2t)-'
iTHi/ vi£N Ti«H mm
A « -fAii.
^^^n
17
Chuy§n dg BDHSG JoAn gia tri Idn nha'l
Tur
(2) suy ra
>
g\& trj nh6 nhS't - Phan Huy Khii
2. D o y > 0, nen
CO
y>
V2
, V X e
Cty TNHH MTV DWH Khang Viet
[-2; 5]
V i the ta C O ( - x ' + 3x + 10)(2x - 4 ) ' - ( - x ' + 4x + 21)(2x - 3 ) ' = -51x^ +
(3)
104x - 29 va c6 bang xet da'u sau:
(a + 3)(5 - x) = (x + 2)(7 - x) <=> x = j .
Da'u bang trong (3) xay ra o
1
87
X
Tuf do suy ra miny = ^/2
cs> x =
N e u f ( x ) > 0 V x e D , thi m i n f ( x ) =
/minf^(x).
-
0
y
PhU'dng phap bat dang thiJc ap dung d day tuy ddn gian nhU"ng c6 hieu qua
Nhifvaytaco
Idn. Theo chiing toi do la each giai hi^u hieu nhat doi v d i b a i loan nay.
1
+
(1)
/g
21.V-x^ + 3x + 1 0
.c
om
Tilf (1) suy ra da'u ciia y ' la dau eua tuf so.
T = (4 - 2x) V-x^ + 3x + 1 0 - (3 - 2x)V-x^ + 4x + 2 1 .
ok
1 |
bo
fa
< - thi 4 - 2x > 0; 3 - 2x > 0,
2
'' '
w.
X
khi - <
X <
2 thi 4 - 2x > 0; 3 - 2x < 0.
IOV2
7A/2
3
3
1
+
+
1l-
1 'tU
/• \1»'.M J -
. . .
BM 5: (De thi tuyen sink Caoddnf^khoi
A-2010)
i
Cho hai so' thU'c difdng x, y thoa man dieu k i e n 3x + y < 1.
1 1
T i m gia t r i nho nhat cua bieu thtfc A = — + - 7 = .
X
^xy ,
' ^
I > •
,
,
Vxy <
•
x+y
,
,
1
1
, do do A = - + -
7 =
2
1
>
-
+
.
(1)
2
X
^xy X x + y
Da'u bang trong (1) xay ra o x = y. L a i theo ba't dang thuTc Cosi, ta c6:
4
1
2
^
2
u
1
2 ^
-X + x + y >2 ^ x ( x + y ) hay -x + x + y > ^ 2 x ( x + y )
3
"
T r d n c d s6 66 ta c6 ngay T > 0 k h i - < x < 2, va
Da'u b^ng trong (2) xay ra o
( 4 - 2 x ) V - x ^ + 3x + 1 0 - ( 3 - 2 x ) V - x ^ + 4x + 21 k h i - 2 < x < 2
1
- =
X
(2)
2
o 2 x = x + y o x = y.
x+y
L a i theo ba't dang thtfc Cosi, ta co: yjlxix
+ y) <
= ^—
(2x - 3)V-x^ +4X + 21 - (2x - 4)V-x^+3x + 10 khi 2 < x < 5.
Difa vao nhan x 6 t sau neu A > B > 0 t h i A > B o
>
,
Hitdng ddn giai
/—
,
ww
.
+
+
Theo bat dang thiJc Cosi, ta c6:
ce
Ta nhan thay k h i x > 2 t h i 4 - 2x < 0; 3 - 2x < 0,
tinh toan t h i qua phiJc tap).
(2)
Chti y r k n g h a m s o y x d c d i n h v d i - 2 < x < 5 .
khi
5
Ta
up
2V-x^+3x + 10
+ 4x +
2
luan cho ban doc (ve nguyen l i thi rat ddn gian chi can x e t da'u y ' , nhU'ng
(4 - 2 x ) V - x ^ + 3 x + 10 - (3 - 2 x ) 7 - x ^ + 4 x + 21
2yJ-\^
51
V e tinh hieu qua cua hai phiTdng phap trong hai each giai x i n danh phan binh
-2x + 3
V-x^ + 4 X + 21
min y = y .3.
s/
2
ro
Ta c6: y ' -
+
2
Ta thu lai ke't qua tren.
phdp chieu bien thien ham so)
- X
87
— — 1
-2
2. Ta hay so sdnh each giai tren v d i each giai khae sau day (suT dung phiTdng
3
+
+
0
0
y'
ddn gian ve can thiJc da hoc kT d cap 2.
1
1.^^^-^^^
1'
3
T
V i e c phat hien ra (2) la dieu binh thiTcJng v i chi dung eac phep bie'n d o i
T =
0
iL
ie
uO
nT
hi
Da
iH
oc
01
/
X
stjr dung tinh chat cd ban sau nay:
'
+
TCf do ta C O bang xet da'u sau:
1. V i e c ph^t hien ra y > 0 V x e [-2; 5] la le tif nhien va dcfn gian. Tir do ta
f \.
i
-51x^ + 1 0 4 x - 29
Binh luan:
Tiif do thay vao (2) va c6: - +
X
x+y
> — ^
3x + y
(3)
•
Cty TNHH ft/ITV DWH Khang Vigt
:huy6n 6i BDHSG ToAn g\i lr| Idn nhS't va gi^ tri nh6 nhS't - Phan Huy Kh4i
Da'u bang Irong (3) xay ra <=> 2x = x + y <=> x = y.
max 1(1) = max
Tir(l)(2)(3)suyra A >
—
(
4
j 1(0); r
M^l
f,^ 251 25
)• = max < 12; — ^ = —
)
3x + y
Thco gia ihic't ihi 0 < 3x + y < 1, nC-n lir (4) co A > X.
Dau bang trong (5) xay ra
191
NhiT the minS = — dal difdc
16
dong th(li c6 da'u bang trong {1), (2), (3) <=> x = y.
o
+ y
NhiTvay m i n A = X<:::>x = y = - .
4
=l
o
x + y =1
i
x - y
3x
(5)
X= y = —
V
4
•
'
2-V^
X =
Jai 6: (De thi tuyen s'mh Dai hoc Cao dcfni^ kiwi D)
X
1
— <=> i
4
Bai 7: (Dc thi tuyen sinh Dai
HUiiitg dan giai
Cho
'-
'
:
,
.»
r(t)
i
-
0
ww
16
w.
1
I
I'd)
2
16
l(t)
1'
1_^
U6.
191
16
1
1
25
A>
-(X-
4
'
=
-{xUy')
+ ^(x-
+ y - - 2 ( x ^ + y - ) + 1.
(1)
(2), va dc y den (1) ta c6
^ t,^
+y-)- + - ( x - + y ) - -2(x- +y") + 1
2
i.u,»,,.
(3)
A> -(x^+y-)--2(x^+y-)+l .
1
4
+
j,'
A p dung bat dang thi?c x"* + y'' >
bo
,
ce
'
fa
,
V i c t lai A d i / d i dang A = ^ ( x ' * + y ^ ) + | ( x ' ^ + y-*) + 3 x - y - - 2 ( x - + y " ) + l
(x^ + " )"
d day l"(t) = Kit" - 2t + 12, vdi 0 < t < ^ . Ta c6 r'(t) = 32t - 2 , va c6 bang
bicn thicn sau:
HUiing ddii giai
'
ok
•
so ihiTc x, y lhay ddi thoa man dicu k i c n (x + y ) ' + 4xy > 2.
.c
min I X t ) ,
om
Dox>();y>();x + y = 1 ncn{)
4
4
-
Ccic
/g
ro
-
()<(<'
^2"2J
Cao dan}' khoi B)
IJOC
s/
(1)
up
Do X + y = 1, ncn la CO S = 16x'y' - 2xy + 12.
(i
<:=>(x;y) =
T i m gia Irj nho nha't ciia bicu ihiJc A = 3(x"' + y ' + x ' y ' ) - 2(x- + y ' ) + 1.
= 1 6 x ' y ' + I2(x + y ) | ( x + y)" - 3 x y | + 34xy.
max f ( t ) ; m i n S =
1
Ta
Ta CO S = 16x'y' + 12(x'' + y ' ) + 34xy = Ifix'y- + 12(x + y)(x" - xy + y") + 34xy
VaymaxS=
+y =1
giiii bai loan nay.
nho nha't ciia b i c u thuTc S = (4x- + 3y)(4y- + 3x) + 25xy.
...
.....W^M
liiiih liigii: Ro rang phufitng phap chicu bicn thicn ham so la thich hdp nhal dc
Cho a i c so thifc x. y khong am va thoa man x + y = 1. T i m gia tri Idn nha't va
Dat xy = I.
4
4
1
noi trcn.
+ S
«
25
lihih liiqn: PhiTdng phap ba't dilng thu'c to ro siJc manh ciia no trong lc(i giai
V a y min
()
2
4
1
xy = —
16
16
.
X =
iL
ie
uO
nT
hi
Da
iH
oc
01
/
Da'u bang Irong (4) xay ra o
Da'u bhng trong (3) xay ra o
dau bang trong (2) xay ra o
x ' = y <=>
Do (X + y ) - > 4xy, ncn lir gia ihict (x + y)' + 4xy > 2, suy ra f>^\
(x + y ) ' + (x + y ) ' > 2 - » ( x + y ) ' + (x + y ) - - 2 > ( )
(X
Do
(X
v
+ y - 1 )|(x + y ) ' + 2(x + y ) + 2] > 0
(4)
+ y ) - + 2(x + y) + 2 = l(x + y) + 11- + I > 0, ncn lir (4) c6:
x + y - 1 >()hay x + y > 1
>
;
4
D a l I = X - + y-, Ihi do x ' + y ' >
ncn itr (5)
2
CO
I> ^.
2
21
Cty TNHH MTV DWH Khang Vigt
Chuyen dg BDHSG Join gia trj I6n nha't \ii g\A trj nh6 nha't - Phan Huy KhSi
nha't va nho nha't cua h a m so', l / u the cua phiTdng phap sijr dung bat d i n g thufc
Tif d6 suy ra \6t h a m so f(t) = ^ t ^ - 2t + 1 v d i t > ^ .
ro net qua thi du nay.
' - ^ ' ;
(De thi tuyen sinh Caoddn}> khoi A)
T a c 6 f ' ( t ) = - t - 2 > 0 V t > - . N h i r v a y m i n f ( t ) = f'
v2.
16
• ii-; ,v > * > ;
Cho x, y la hai so thi/c thoa m a n x^ + y^ = 2.
T i m gia trj Idn nha't va nho nha't cua bieu thiJc P = 2(x^ + y^) - 3xy.
TH do suy ra (diTa vao (3)): A >
(6)
D e thay dau bang Irong (6) xay ra
1
o
x=y
9
1
N h i / t h e ta CO m i n A = — o x = y = —.
16
^
Hudng dan giai
iL
ie
uO
nT
hi
Da
iH
oc
01
/
16
o x =y=—
1
t = 2
Ta co: P = 2(x + y ) ( x ' + y ' - x y ) - 3xy = 2(x + y)(2 - x y ) - 3xy.
(1)
Ta c6: x ' + y^ = 2
(2)
(x + y)^ - 2xy = 2
xy = ^ ^ ^ ^ ^ ~ ^ •
Thay (2) vao (1) r o i dat t = x + y, ta c6:
P = 2t 2 -
2
.2
_ 3 ! l _ l =-e-^t^ + 6t + 3.
t^-2^
• A
7.
Bay g i d ta t i m m i e n xac dinh cua t:
Bi?ih ludn: V i e c diTa ve xet bieu thtfc d o i v d i x^ + y^ la mot le tiT nhien.
Bang phu'dng phap bat dang thtfc ta di den danh gia (3).
2
2..,
2 = x' + y >
,2
(x + y )
— —
>(x + y ) ^ < 4 c : > - 2 < t < 2 .
Ta
Cong viec con l a i di nhien can den viec van dung phiTdng phap chieu b i e n
, , .«nU;3w,;;:M''..^J.-
thien ham so'.
s/
B a i toan quy ve: T i m gia tri Idn nha't va nho nha't cua ham so:
ro
chieu bien thien ham so da dan den siT thanh cong trong qua trinh giai bai
up
Ta thay viec ke't hdp nhuan nhuyen giiJa hai phifdng phap bat d i n g thiJc va
/g
toan nay.
Bai 8: (De thi tuyen sinh Dai hoc Cao ddn^ khoi B)
-2
.c
1
i
f'(l)
ok
^^^^
l + 2xy + 2y^
Ta CO f ' ( t ) = - 3 t ^ - 3t + 6 va CO bang bien thien sau:
t
bo
T i m gia t r i Idn nha't va nho nha't cua bieu thtfc P =
'
om
Cho X , y la cac so' thifc thoa man dieu k i e n x^ + y^ = 1.
f(t) = _ t 3 _ | t ' + 6t + 3 v d i - 2 < t < 2
f(t)
1
^
0
~
T
_
\
fa
X e m Idi g i a i (ba each) trong bai toan 3, muc § 1 , chiTdng 1 cua sach nay.
w.
Binh luqn: B a i toan c6 den 3 Idi giai khac nhau (hai each suT dung phiTdng phap
VaymaxP=
ww
each suf dung phu'dng phap chieu bien thien ham .so').
minP=
-i"
,
T i m gia t r i Idn nhat va nho nhat cua bieu thiJc P =
f ( t ) - f ( l ) = •5:^;
2
m i n f ( t ) = m i n { f ( - 2 ) ; f ( 2 ) } = m i n { - 7 ; 1} = - 7
-2
B a i 9: (De thi tuyen sink Dai hoc Cuo ddn}^ khdi D)
Chox>0;y>0.
max
-2
m i e n gia t r i h a m so trong do c6 ke't hdp ca phiTdng phap li/dng giac hoa, mot
:
(x-y)(l-xy)
(l + x ) ' ( l + y ) '
X + y = -2
minP = - 7 <=> <^ 2 , . . 2
_x^+y^=2
X
Hudng dan giai
X e m Idi giai (bon each) trong bai toan 2, muc § 1 , chtfdng 1 cua cuon sach nay.
Binh luqn: D a y la m o t trong cac bai toan the hien ro nha't tinh da dang cua viec
suT dung cac phu"dng phap khac nhau de giai mot bai toan ve t i m gia t r i Idn
-<^^=y—h
• ^ \-S
maxP = —
2
1
2
ce
Hitdng dan giai
^,, , ^
x = = — ; y =
+y =1
l + ^/3•
—
o
x2+y^=2
;
X
=
l + ^/3
1-N/3
-;y = -
Cty TNHH MTV DWH Khang Vigt
Chuy6n 6i BDHSG loin g\ tri I6n nhS't va gia tri nh6 nha't - Phan Huy Kh^i
v i i C O b a n g b i c n i h i c n sau:
I
t)
r'(i)
1
1. V i c e quy v c b i c u ihuTc d o i v d i x + y ( m a la sc d a i b a n g I ) r o i stir d u n g
p h i f d n g p h a p e h i c u b i c n i h i c n h a m s o ' d c g i a i la v i c e l a m u / n h i c n ( g i o n g
1
0
-
nhi/ irong nhicu bai loan x c l tri/c'tc day)
i'(i)
2. Ta ihur suy n g h l d i c u k i c n x~ + y " = 2 g(;i y la eo n c n siir d u n g p h i f t t n g p h a p
" l i M n g g i a c h o a " hay k h o n g ?
V a y m i n 1(0 = 1(1) = ^
yfl sincji; y = sfl cos(p
L i i e n a y P CO d a n g sau:
,
.
iL
ie
uO
nT
hi
Da
iH
oc
01
/
V l X" + y " = 2, n c n d a i \=
>
•
, ,
,
, ,
TCr do suy ra V
P = 4 v'2 ( s i n \ + c o s \ p ) - 6sin(peos(p
(sincp + cos(p)( 1 - sinci^coscp) - 6sintpcos(p
ro
so lU"(tng ur n h i r d a l a m I r o n g p h a n I r c n . Vay sir d u n g " l i r d n g g i a c hoa " c h i
/g
la p h a n d a u , c o n l i c p i h c o la i a i si? d u n g phiriJng p h a p c l i i c u b i c n ihiC-n
om
h a m so n h i r d a l a m ('UrcMi.
.c
Dai line (\ii> c/(?/;,^' klioi B)
/.
I
HUi'Stifj dan fiidi
Ta
CO
S =
V"
1
/"
X"
•+— + •
-)
• \
+
XV/
H i c n n h i c n la c6: x ' + y " +
> xy + y / + / x .
D a u b i u i g I r o n g (2) x a y ra
x = y = /.
T i r ( l ) (2) la CO S >
'x^
1
— +—
v 2
\
/••" ; f . . v - . ,
+ 2
I
(3)
2
y; ,
/.
2
(4)
Mi
> 0. y > 0, / > 0.
; ;,
, , ; '
,^
<=> x ' - 3x + 2 > 0 <=> (x -
o
Tir do suy ra S > -
l)(x'
.
. :
''
(5)
2) > 0
(6)
(5) dung V X > 0.
=> minS = - < = > x = y = / = l .
X c m ra each giai nay qua ddn giiin. Nhu"ng
(2)
+ x -
(X - i r i x + 2) > 0.
R6 rang (6) diing Vx > 0
(I)
irJi
g i i i i ciia no khong ur nhicMi
& cho: lam sao ma b i c l difclc vc phiii ciia (5) la ^ .
^^y\2i
{De thi tuyen sinh DiiUwc
Cci() ddn}> khoi A)
Cho X, y la cac so ihirc khac 0 va ihoa man d i c u k i c n xy(x + y) = x ' - xy + y"I
(3)
+—
T i m gia i r i Idn nhal ciia bicu ihiVe: A =
+
^
1
X c l ham so 1(1) =
T h c i t vay
ce
/
2
x'
I
3
nay. T h i l l vay la sc chi^ng minh rang k h i x > 0, Ihi — + — > — .
2 x 2
fa
yy
+ y —+ — +
V 2
/,x )
y
2. Ta t o Ihc siir dung ihuan liiy phu'ctng phap ba'l dang ihii'c dc g i i i i bai loan
w.
[2
]
ww
I
2
1 3
phuTilng phap c h i c u b i c n i h i c n ham so cho phcp la giiii bai knin i r c n .
bo
Cho X, v, / la ba so dirt
S= x
2
3 /.^
1. M o l Ian nffa la lhay sir k c l hdp giffa phiTdng phap ba't d^ng ihiJe vii
ok
B a i I I : {De thi ttiycn
x
I
Ta
K h i d o la h i i q u y v c b i i i l o a n l i m gia l i i ii'Jn n h a l va n h o n h a l ciia m o l h a m
:
X
3 y^
Gia irj nho nhal da I du'dc k h i va chi k h i x = y = / = I .
r -1
-6
V
—
1
V a y minS = ^ .
< l < V2
s/
la C O P = 4N/2I 1 -
-1
> 0, y > 0, /. > 0, la c6:
X~
Dau bang Irong (4) xay ra <=> x = y = z = 1.
up
I-
(do do -\[2
X
TiT (3) suy ra S > -
.,
.
, •
(sincp f cos(p)" - 1
A p d u n g c i ) n g Ihifc smtpcoscp = —
va d i l l I = sincp + eosip - V2 cosj
l>0
vi'fi l > 0. Ta ci) I ' d ) - 1 -
I-
HUdiig dan giai
I ' -1
— r
Do x
7^ 0 ,
y
;t 0
n c n xy(x + y) = x ' - xy + y"
•
y
:
•
Cty TNHH MTV DWH Khang Vi$t
Chuy6n dg BDHSG Jo&n gii tr| I6n nhflt
1 1 1
O
1
1
- + - = — + —
X
g\i trj nh6 nha't - Phan Huy Khi\
.
.
2y
.
y
'
xy
(1)
^.
• i
D a t X - - ; Y = - k h i do (1) c6 dang: X + Y =
X
y
- XY + YI
f'(y)-
(2)
V d i dieu k i e n (2) t h i A = ( X + Y ) l
Vl2y>
T i m gia trj Idn nha't cua A = (X + Y)^ v d i dieu k i e n (2).
X + Y = (X + Y)^ - 3 X Y .
'
.
;„
^4J;'
^
y
r
( X + Y)^
nen tuf (4) ta c6
(X + Y)^ - 4(X + Y ) < 0 o
V a y t a c o A < 16.
-
•
(5)
I'..'
.E^-f,
+7(x + l ) ^ + y ^ + y - 2
.c
HUbng dan gidi
Theo phep tinh ve vectd ta c6:
ok
L a y i i = (x - 1 ; y ) , V = ( - x - 1 ; y) => u + v = (-2; 2 y ) . '
w.
Da'u bang trong (1) xay ra <=> u, v la hai vectd cung phi/dng, ciing chieu
ww
1 = - X - 1 <=> X = 0.
T i r ( l ) s u y ra A > 2^1+ y^ +|y-2|.
9fi
up
Nhi/vayA>2+ N/3VX, y G R v a A
2^/r+y^ + y - 2
neuy>2
2sjl + y^ + 2 - y
neuy<2
;
Binh luqn: R i n g phep tinh vectd ta c6 danh gia (1). Sau do suf dung phiTdng
tren doan [ 1 ; e^].
X
•(2)
'
v
T i m gia t r i I d n nha't va nho nhat cua h a m so f(n) =
v
•
. ,.V?*V -
= 2+73c>-
Tilf do ta CO mina = 2 + N/3<:=>X = 0, y = ^ .
Hi/dngdan
x21nx.i-ln'x
Tac6 f'(x) =
vdiyeR.
E.
x = 0
• '
Dau b^ng trong (2) xay ra <=> X = 0.
X e t h a m s o f(y) = 2 7 l + y^ + y - 2
Vye
B a i 1 4 : (De thi tuyen sinh Dai hoc va Cao ddnfi khoi B)
ce
(1)
fa
+ l)2+y2 > 2 ^ 1 ^ ^
V3
phdp chieu b i e n thien h a m so de g i a i tiep bai toan dat ra.
bo
u + V hay
'=2+
ro
? I •
om
V ( x - l ^^+y^
)'
.'
/g
Cho X, y e R. T i m gia tri nho nhat cua bieu thuTc:
Ta CO f(y) =
+
+
Ta
. •vj..^.,
B a i 1 3 : (De thi tuyen sinh Dai hoc Cao ddnjf khoi B)
X -
0
TuT do suy ra f ( y ) > f
s/
de giai bai toan nay
o
^/^7ne'u:^
r(y)
0 < X + Y < 4
B i n h luan: R6 rang phu'dng phap baft dang thiJc suT dung d day day hieu qua
^ ( X - l ) 2 + y 2 +J(x
• ne'u y < 2
y
;\ ••• ' f
•
(3)
f(y)
c ^ ( X + Y ) ^ < 16.
A=
i + y'
ne u y < 2
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01
/
K h i do bai loan da cho t r d thanh:
X + Y > ( X + Y)^ - - ( X + Y)^
4
1 ne'u y > 2
r +
Ttf (3) suy ra f ' ( y ) > 0 V y > 2.
(3)
A p dung bat dang thtfc quen bie't X Y <
r-1
Jy^
neu y > 2
+ y^
L u c nay A = X ' + Y ' = ( X + Y)(X^ - X Y + Y^).
Chu y rang (2) o
2y
.+ 1
giai
•
21nx-ln'x
lnx(2-lnx)
x^
X'
D o x^ > 0 V X e [ 1 ; e^ nen ta c6 bdng bie'n thien sau (di/a vao tinh dong b i d n
cua h a m so y = Inx k h i X > 0)
Cty TNHH MTV DVVH Khang Vi?t
X
1
Inx
i
i
2 - Inx
+
y'
+
0
_
+
0
-
L a i CO ! ' ( - 1 ) = 0. Tu" do suy ra
—
1
...
1
Vcjy max R x ) - ("(c") = 4";
niin
min !(x) - 0 .
•
<-2. > (X + 1)"
X- + l + 2x
,
2x
Laiihay r ( x ) =^
=
= \ —rx- + 1
X- + 1
x^+1
r
9
0-^
> ( ) V x e 1 - 1 ; 2|.
,
4
2x
Dol
< 1
f ' ( x ) < 2 =5. l"{x) < N/2 , V x 6 1 - 1 ; 2|.
iL
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/
-' y
1
+
x+ 1
D o x > - l = ^ x + 1 >()=>r(x) =
x^ + 1
M a i khae 1 ( 1 ) = %/2 , iCr do suy ra
l"(x) = m i n i 1(0); r(c^)} = m i n j o ; ^1 = 0.
max H\) = \f2 .
-I
3. L a i CO Ihc su" dung phiidng phap lu'dng giac hoa nhU sau:
liinh ludit: PhiTdng phap c h i c u b i c n Ihicn ham so la each g i a i duy nhal hdp l i
v d i bai loan nay.
\
.
•
B a i 15: (De thi tuyen sink Dai hoc Cao cldiif^ kiwi D)
D o - 1 < X < 2, n c n dal x = lancp v d i - — < (p < arelan 2 .
4
:
v
'••
2 2
Ta
:
Hit(yiig ddn giai
Ta CO
1 +x
-J
1
2
bo
1
.c
L
y'
ok
X
1
ce
S
w.
i
fa
y
0
ww
'MM,
-l
71
Til ~ — < cp < arelan2 => — - arelan 2 < — - cp < — . (2)
;
4
4
4
2
•
.••>•,•.
i,i';>'i.; •:• .•• •
NhiT vay vciti chu y rang — < — arclan2 < 0 , n c n luT (1) (2) suy ra
4
4
1+x
m i111
n
,
-i< x . 2 ^ 1 + X^
= \/2 min cos - - c p
= 0 k h i cp = — ,
V
1+X
-l
max cos
-cp
= V2 k h i cp = 0 .
1 + x^
D i e u dc) c6 nghla la minr(x) = 0 <=> x = - 1 ; max r(x) = 72 <=> x = 1.
R6 rang trong ba each ke Iren phufclng phap suT dung chieu bien ihien ham
fii/j/i /ua/i;
1. V d i I h i d u i r e n suf dung phU'dng phap chieu b i c n i h i c n h a m so de g i a i b a i
loan la h(fp If nhal.
1
= V2eos - - c p
max
-- lK< x'
eoscp
= smcp + eoscp
om
V a y la eo bjlng bicn thicn sau:
_
cos" (p
up
1-x
ro
Vx^+1_
x^ + 1
/g
Tae6r(x) =
1 +lancp
sm(p + cos(p
+ lan" 9
s/
V
=> eoscp > 0.
Chu y rhng khi do cp e
x+ 1
T i m gia I n k'ln nhal va nho nhal eua ham so l ( x ) = .
Ircn doan ( - 1 ; 2].
... . .
V x ^+ •1
^•r...,^-;:,t^,f;,<^,':r-y•
2. Ta CO Ihe siif dung phi/dng ph
so irong each 1 la hieu qua va hc;p l i nhal.
B a i 16: (De thi tuyen sinh Dai hoc Cao dang khoi B)
T i m gia Irj k?n nhal va nhc) nha'l cua ham so f(x) = x + yJA-x^
trcn m i c n |
xac d m h cua no.
. .
29
Cty TMHH MTV D W H Khang Vi$t
Chuyfin dg BDHSG loan g\i trj Idn nhat va gia trj nh6 nhift - Phan Huy KhSl
HUdng ddn gidi
Tir do suy ra P > 8, vay minP = 8.
Xem Idi giai trong bai loan 1, muc §1, chiTdng 1 cuo'n sach n^y.
Cach giai nay sai d ch6 la mdi difa vao phan 1 cua dinh nghTa gid tri nho
nhat. Ta xem phan 2 cua dinh nghTa c6 thoa man hay khong? De y r i n g da'u
giai khac nhau dung de giai mot bai toan tim gia trj Idn nha't va nho nha't cua
bing xay ra trong bat ding thuTc P > 8 k h i x = y = l . Tuy nhien khi d6 x^ + y^
ham so (bai toan c6 den 4 each giai khac nhau suT dung hau he't cac phiTdng
= 2 (khong thoa man dieu ki$n x^ + y^ = 1). Vay khong the xay ra da'u bing
phap cd ban de tim gia trj Idn nhat va nho nha't cua ham so: phiTdng phap bat
trong baft d i n g thtfc P > 8, tuTc la phan 2 cua dinh nghTa ve gia tri nho nha't
dang thuTc, phi/dng phap chieu bien thien ham so, phiTdng phap liTdng giac
khong diTdc thoa man.
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Binh luan: Day la mOt trong cac vi du dep nha't ve tinh da dang cua cdc each
hoa, phtfdng phap mien gia tri ham so, phiTdng phap do thi va hinh hoc).
.
4 Hi
Vi thS' ket luan minS = 8 la sai.
(Hi
SI
Cdch giai dung nhif sau: Viet lai S diTdi dang:
§3. C O s d LI THUYETCUA BAI TOAN
1 x 1
y
S=l+x+-+-+l+-+y+y y
X
X
TIM GIA TR! LdN NHAT VA NHO NHAT CUA HAM SO
f
A. Dinh nghia gid tri Idn nhat va nho nhat cua ham so.
I
Dinh nghTa 1 : Xet ham so f(x) vdi x e D. Ta noi rang M la gia tri Idn nha't ciia
f(x) tren D, neu nhiT thoa man cac dieu kien sau:
1. f ( x ) < M , V x 6 D,
tai
Xo e
D,
sao
cho
f(X()) =
M.
up
Khi do ta ki hi$u: M = max f ( x ) .
ro
XfcD
/g
Djnh nghTa 2: Xet ham so f(x) vdi x e D. Ta noi r i n g m la gia tri nho nha't cua
om
f(x) tren D, neu nhuT thoa man cac dieu kien sau:
tai
X ( ) 6 D,
sao
cho
f ( X ( ) ) = m.
•(
ok
Ton
.c
1. f ( x ) > m V x e p ,
2.
—
+ 2.
(2)
>J2;
(3)
ce
xeD
,'ff.,ftu
Matkhac - + - > - ^
X
y Vxy
Do
X
1 1
+ y^ > 2xy, nen tir (4) ta c6 - + - >
X
y
- 1< 5"
bo
Khi do ta ki hieu: m = min f ( x ) .
Tir(l)(2)(3)va (5)suyraS>3N/2 +4.
• = 2V2.(5)
V
x2+y2
(6)
Dau bing trong (6) xay ra <=> dong Ihdi c6 dau bing trong (2), (3), (5)
fa
Nhir vay dinh nghTa gia tri Idn nhat va nho nha't deu c6 hai phan. Can liAi y
(1)
yj
Ta
Ton
Tac6: x + — >>/2; y + —
2x
2y
. .
s/
2.
,
1
X
y
1' f
1 '
x + — + y+zr- + - + - +— — +
2 y ; ly ^ ) 2 U
2x> [
w.
r i n g ca hai phan nay deu quan trong nhu-nhau, khong diTdc xem nhe phan 2.
o x =y=
ww
Xet thi du sau day:
Cho X > 0, y > 0 va x^ + y^ = 1. Tim gid tri nho nhat cua hiiu thtfc sau:
Nhir vSy ton tai (x,,; y«) thoa man xf, +y?) = 1 va S = 3^12+4 khi x = x,,;
P = (l +x)
y = y,). Theo dinh nghTa ve gia tri nho nhat, ta c6 minS = 3yl2 + 4 .
+ (l +y)
yj
Xet phep giai sau day: P = 2 + f X
I
Qua thi du nay ta thay neu khong de y den di^u kiC*n 2 trong dinh nghTa gid
1^ +
x;
+ —
f y + 1^
- +
k
X
y
— + -
tri Idn nhat v^ nho nha't cua ham so c6 the se dan den sai lam.
y)
Taco x + - > 2 ; y + - > 2 v a - + ^ > 2 .
X
y
y X
31
Cty TNHH MTV D W H Khang Vigt
Chuy6n dg BDHSG ToAn gia tr| Mn nhift vji glii tr| nh6 nha't - Phan Huy KhJi
B. Cdc tinh chat cua gid tri l^n nhat vd nho nha't cua ham so.
min f(x) = min min f(x); min f(x)|>.
Gia sur ton tai max f(x); max f(x); min f(x); min f(x).
xeA
Khi do ta
CO
xeB
xeA
i : . > > ;•
max f(x) < max f ( x ) ,
(1)
min f ( x ) > m i n f ( x ) .
(2)
xeA
xeB
xeA
Chtfngminh:
v ;
xeB
xeB
suf max
xeD
f(x) = f ( X ( , ) , vdi x,, e A. Do X(, e A ma
A c
.
xeB
•
•'" « '
'
toSn tifdng lir). V i Di c D, I = 1, 2 nen theo tinh chat 2, ta c6:
xeDi
X(, e
Gia
t
•" *
< ,
;
xeD
xeD
Chi?ng minh: Gia suf max g(x) = g ( X ( , ) , vdi x„ e D.
Ta CO f(x)
> g(x)
V X e D =^ f(xo)
V',
v..
up
Do max f(x) > f(x,,) > g(x„) = max g(x) => dpcm.
xeD
ro
xeD
.c
thi chi/a the kct luan diTdc f(x) > g(x) V x e D.
xeD
xeD]
xeDi
ww
Xet mot vi du minh hoc sau day:
D2.
Chox>0, y>Ova x + y < 6
t^.
Tim gia tri Idn nha't cua bieu thiJc P = x^y(4 - x - y).
DatD= {(x;y):x>0;y>0; x + y<6},
Khi do ta c6 cong thtfc sau: max f(x) = max \x f(x); max f(x) L
^7
1
VI11 do ay tinh chat 3 con hay gpi la NGUYEN L I PHAN RA.
xeD;
["^'^1
xeD2
mien ddn gian hdn (cac bai toan ay ddn gian hdn cac bai toan ban dau)
Gia thiet ton tai max f(x), min f(x) V I = 1, 2.
xeD
(7)
xeD2
day cac bai toan tim gia trj Idn nhat, gia tri nho nha't cua ham so tren cac
Nhir vay khong the suy ra duTdc f(x) > g(x) V x e D..
xeD:
Tif (5), (6) suy ra f(x„) = max f(x) < max max f(x); max f(x)
gia tri nho nha't cua mot ham so tren mot mien xac dinh phtfc tap thanh mot
' Tuy nhien vdi mpi 2 < x < 3 thi ro rang f(x) < g(x).
f(x) xdc dinh tren mien D va D = Dj u
xeD2
Nhd tinh cha'l 3 noi tren cho phcp ta c6 the bien bai loan tim gia tri Idn nhat,
w.
max g(x) = 3.
SIJT
(6)
[xeDi
=> Do la dpcm.
fa
()
Tinh chfi't 3: Gia
Hien nhien max f(x) < max< max f(x); max f(x)}-.
xeD
ce
R6 rang max f(x) = 4 ,
()
(5)
xeD|
Bay gid tir(4), (7) di den: max f(x)=:max-^max f(x); max f(x)
bo
tren mien D = {x: 0 < x < 3 } .
D.
Tif Xo e D | nen theo dinh nghia ve gia tri Idn nhat, ta c6:
f(x„)
ok
Xet hai h^m so f(x) = 4 - x va g(x) = x
0
max f(x) = f(x„), vdi Xo e
om
/g
Chii y: Menh de dao noi chung khong dung, ttfc la neu max f(x) > max g(x)
Nhifvay max f ( x ) > max g(x). S
siJf
xeD
xeD2
quat)x„GD,.
xeD
> g(Xo).
,
•
mot trong hai tap D j , D2. Tif do c6 the cho la (ma khong lam giam sur tong
'
xeD
^
VI D = D i U D^ma X,)G D nen x,) e D, u D2. Do vay x,, phai thuoc ve it nhat
«j -
cijng ton tai max f(x); max g(x), khi do ta c6 m a x f ( x ) > m a x g ( x ) .
xeD
(3)
xeD
xeD
Ta
SIJT
xeD2
[xeDi
s/
Gia
x^t)
,^
Tif (3) suy ra max-^ max f(x); max f(x) < m a x f ( x ) .
B.
Tinh cha't 2: Gia sur f(x) va g(x) la hai ham so cung xdc dinh tren mien D va
thoa man dieu kien f(x) > g(x) Vx e D.
-
niaxf(x)
B nen
Taco f ( x „ ) < m a x f ( x ) =>dpcm.
.
(2)
xeD2
Ta chi can chu'ng minh (1) (con (2) diTdc chi^ng minh b^ng mot each hoan
Chtfng minh: Ta chlJng minh (1) (con (2) chtfng minh hoan toan tiTdng tif)
Gia
,
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i
xeD)
xeD
Tinh chfi't 1: Gia siJ f(x) xdc dinh tren D va A, B ia tap cua D, trong do A e B.
"^'^2
(1)
'h
D | = {(x; y): x > 0 ; y > 0 ; 4 < x + y < 6 } ,
D2= {(x;y):x>0;y>0;x + y<4).
33
Chuyen dg BDHSG Join gJA irj I6n nhat va g\A tr| nh6 nha't - Phan Huy KhSi
K h i do ro rang D = D , u D 2 .
Cho x > 0; y > 0 va X + y < 6.
Theo nguyen l i phan ra (tinh chat 3), ta c6 :
T i m gia t r i nho nhat cua bieu thuTc P = \^y(4 - x - y).
max
(x;y)eD
P = max
max
P;
(x;y)eD|
K i h i c u D , D | , D2 nhif trong thi du minh hoa cua tinh cha't 3.
max P
(1)
(x;y)6D2
Theo tinh chat 2 thi
V d i m o i (x; y ) e D , thi 4 - x - y < 0 v i do x > 0; y > 0 nen P < 0 V (x; y) € D,
L a i c < 5 ( 2 ; 2 ) G D| va k h i d 6 P = 0, ncn
max P = 0.
iL
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Da
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oc
01
/
^+-^- + y + ( 4 - x - y )
y =I
(do khi (x; y)
,
ro
xeD
ChuTng m i n h : Gia suT M = max f ( x ) .
xeD
xeD
,
om
xeD
.c
xeD
/g
min f ( x ) . K h i do ta c6: max f ( x ) = - m i n ( - f ( x ) ) ; min f ( x ) = - m a x ( - f ( x ) ) .
.
ok
X6D
max Q = max{();64) = 6 4 .
(x;y)€D
Bay gid tCr (1), (4) suy ra
,
(4)
, I
mm P = - 6 4 .
O.I'
(x;y)€D
T i n h cha't 5: Cho cac h a m so l"i(x), i'2(x),... f„(x) cung xac dinh tren m i e n D.
D a t t ( x ) = l|(x) + t2(x) + ...+ l„(x).
'
Gia stir ton tai m a x f ( x ) , m i n f ( x ) , m a x f j ( x ) , m i n f i ( x )
X6D
xeD
xeD
xeD
vdi moi i = i . n .
.
K h i do ta c6: m a x f ( x ) < m a x f , ( x ) + maxf2(x) +... + m a x f „ ( x ) ,
xeD
xeD
xeD
xeD
(1)
xeD
xeD
(2)
(
NhiT vay ta d i den max f ( x ) = - m i n ( - f ( x ) ) =>dpcm. ,
maxi;(x) = fi(x„),Vi = l . n .
,
Phan sau chiJng m i n h hoan toan tiTctng tiT.
i
'
Nhan xet: T i n h cha't 2 cho phep ta chuycn bai toan t i m gia tri Idn nhat thanh bai
todn t i m gia t r i nho nhat hoSc ngmJc l a i . D i e u nhy c6 ich trong nhieu tri/dng
hdp cu the se xet sau nay.
X 6 t t h i du m i n h hoa sau day:
1A
max Q = 64 .(3)
Da'u b^ng trong (1) xay ra k h i va chi khi ton tai x„ G D sao cho
xeD
'
'
*
D,
(x;y)eD|
Theo dinh nghia cua gia t r i nho nhat, tCf (*) suy ra m i n ( - f ( x ) ) - -M .
xeD
<64
Tur (2) (3) va theo tinh cha't 3 (nguyen l i phan ra), ta c6 :
xeD
, ...
\
; ,
min r(x) > m i n f, (x) + m i n f2(x) +... + m i n t'n ( x ) .
w.
!-f(x„) = - M .
thi X + y < 6 => X + y - 2 < 4).
xeD
fa
VxeD
ww
TCf he tren suy ra
ce
bo
;M V x e D
K h i do theo dinh nghla gia t r i Idn nhaft, ta c6: • ^^^^ ^
[f(x„) = M,vdix„eD.
f-f(x)>-M
G D,
x +y - 2
V (x; y) e
Ta
,
s/
max P - m a x { 0 ; 4 } = 4 .
(x;y)eD
Tinh cha't 4: Gia suf h a m so f ( x ) xac dinh tren D va t o n t a i max f ( x ) va
xeD
hay Q < 4
Mat khac do (4; 2) G D , va khi do Q = 64, nen ta c6
(3)
up
Tir(l)(2)(3)suyra
(2)
-]2
Q= 4 - - y ( x + y - 4 ) < 4
x =2
max P = 4 .
max Q = 0 .
(x;y)eD2
^- + ^ + y + (x - y - 4)
X X
h a y P < 4 V ( x ; y ) e D2.
(x;y)eD2
(1)
K h i do (x; y) G D | => x + y - 4 > 0, nen theo ba't dang tMc Cosi, ta c 6 :
^4
Ro r a n g ( 2 ; 1) e D2=>
'
(x,y)€D
Mat khac (2; 2) G D2 va khi do Q = 0 nen ta c6
V d i m o i (x; y) e D2 thi 4 - x - y > 0, nen theo ba't dang thufc Cosi, ta c6:
M a t khac lit — - y = 4 - x - y < = >
m i n P = - max ( - P ) .
(x.y)eD
Ta CO - P = Q = xV(x + y - 4). K h i (x; y) e D2 =^ X + y - 4 < 0 => Q < 0.
(2)
{x;y)eD|
P = 4||y(4-x-y)<4
*
xeD
Da'u bang trong (2) xay ra k h i va chi khi ton tai x„ G D sao cho
minf,(x) = fi(xo),Vi = l , n ,
X G (J
ChiJng minh:
Ta chiJug m i n h (1) (vdi (2) phep chilng minh hoan toan tu'dng tiT).
La'y tOy y X e D . Theo dinh nghia cua gia t r i Idn nhat ta c6 :
35
Cty TNHH MTV DWH Khang Vi$t
Chuy6n 06 BDHS6 Toan gia tr| I6n nha't va gi& tr| nh6 nhft - Phan Huy KhSi
fj(x)
(3)
Cong tiTng ve n bat dang ihtfc (3), ta c6: , :
,
^
j ' ^
f(x) = f,(x) + ... + l n ( x ) < m a x r | ( X | ) + ... + maxfn(x). (4)
xjeD
Do t6n tai x„ ^ y , ma g(x„) =
. >
XjeD
>
V i bat d i n g IhiJc (4) dung \6i moi x e D nen ta c6
maxf(x)
eD
,,
, , ,
(5)
.
^
eD
Xj
Vi ic do theo tinh chat 5, ta c6 :
Vay (1) dung. Bay gict ta xct kha ndng c6 dau bang trong (1).
in t'(x)= min g(x)+ min h(x) + 5 = - ^ + 8 + 5 = . ^
mm
kn
kJt
kn
2
2
kn
G i a s u r t o n t a i x i i s D m a maxfi(x) = ri(X()), V i = l , n .
xfeD
T t f d o t a c o maxf,(x) + ... + maxf„(x) = f|(x,)) + ... + f (x„) = f(x,)). (6)
xeD
N6
Tinh cha't 6: Gia siJf f,(x), r:(x),... f„(x) ciing xac djnh trcn mien D, va ta c6
D
Ux) > 0 V X e D, Vi=:i7n. Gia ihict them ton tai maxr|(x).maxl2(x),...
D o f ( X ( ) ) < maxf(x),ncntu'(6)suyra
V
'
xeD
. • ,
maxf|(x) + ... + m a x L ( x ) < m a x r ( x ) .
xeD
xeD
(7)
xeD
xeD
Ta
s/
up
Tinh cha't Ircn cho ta Ihay rhng noi chung khong the thay vice tim gia trj k'Jn
ro
nhat (nho nhat cua mot tdng cac ham so bhng vice tim long cac gia tri Idn
/g
nha't (nho nha't) cua tiTng ham so ddn Ic. Tuy nhicn irong thifc tc dicu nay sc
cos^xy
.c
ok
bo
1
•sin^ x y
J
f(x) = sin''x + cos^ +
sin''x
...
vdi \
kn
— (k e Z)
2
J
"eD
xtD
maxf(x)<
xeD
max l"|(x)
xeD
xtD
niinf(x) > m i n l ' | ( x )
xeD
max r-,(x)
xeD
"
min r-,(x)
xeD
"
>^
j
l--.sm''2x
+ 4 = 1 - -sin^ 2x + — ^
— +4
cos''xy
2
sin xcos x
xeD
'v^^
xeD
Da'u b a n g trong (1) xtiy ra k h i va chi khi ton tai x,)e D sao cho
minri(x) = ("((x,)), V i = l , n ,
, . .;
xeD
Ti/dng tirdau bang Irong (2) xay ra khi va chi khi ton lai x,, e D sao cho
niini;(x) = i ; ( X o ) V i = U ,
'
•
xeD
Chu-ng m i n h h o a n l o a n lifdng liT nhiT chi'fng minh ciia tinh chat 5.
h(x) = f(x) - g(x). Gia stir ton tai cac gia Irj Idn nhat, nho nha't c i i a cac h a m so
f(x), g(x), h(x) trcn D. Khi do ta cd: m a x h(x) < m a x l"(x) - m i n g ( x ) , (1)
m i n h(x) > min f(x) - m a x g(x). (2)
(1)
1 - ' sin^ 2x
XGD
xeD
Dau bhng trong ( 1 ) x a y ra khi va chi khi ton tai x„ e D sao cho
maxiXx)=:l"(Xn);ming(x) = g(x„),
xeD
vdi g ( x ) = — s i n ' ' 2 x ; h ( x ) = 16
2
(2)
niinl„(x)
xeD
J
1 - .sm 2x
=5--sin22x + 16—^= 5 + g(x) + h ( x ) ,
. 2
,
sin^2x
(1)
max r,(n)
Tinh cha't 7: Gia siif l"(x) va g(x) la hai h a m so c u n g x a c dinh i r c n m i e n D. Dill
ww
Viet lai f(x) dxid'i dang sau
/
I
fa
.
w.
2
sm" x +
ce
Tim gia tri nho nha't ciia ham so
+
om
thiTc hien diTdc neu nhiT trong cac trifrJng hdp ton tai mot diem x,, ncn cac
Xet thi du minh hoa sau day:
D
X6
Dat f(x) = fi(x)r:(x) ...l'„(x). Khi do la c6:
Phan dao chiJng minh hoan loan imtng lif va xin danh cho ban doc.
ham thanh phan ciing d;U gia trj Idn nhat (nho nha't) tai diem a'y.
xeD
maxf„(x) Cling nhu" minl'ifx),... minf„(n).
f
Tiif (5), (7) suy ra trong triTdng hdp nay xay ra dau bang trong ( I ) .
l(x) = cos' X
/(•A
1
va h(x„) = 8 (thi du chon x„ =
ttfc la ton tai x„ ma g(x„) = mini g(x); h(x„) = min h(x)
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X6D
(2)
DC-thay ming(x) = - ^ ; min h(x) = 16.^ = 8 .
xeD
•
xeD
Dau b a n g trong (2) x a y ra k h i va chi khi Ion tai x„ e D sao cho
sin" 2x
37
Cty TNHH M T V DVVH Khang Vjgt^
Chuyfin 66 BDHSG Toan gia t r i Idn nha't va gia trj nh6 nha't - Phan Huy K h i i
minf(x) = f(X(,);maxg(x) = g(x,)),
xeD
;
xeD
•',
2. Neu them vao gia thie't f(x) > 0 V x e D. Khi do vdi moi n nguyen dtfdng,
,
ta co: max f(x) = 2n/max f^"(x); m i n f ( x ) = 2n/min f^"(x) .
C/ji^n^'m//?/j.-Ta chi can chi^ng minh (1).
xeD
Ta CO h(x) = f(x) - g(x) = l(x) + (-g(x)).
Theo tinh chat 5 ta c6: maxh(x)
xeD
xeD
thtfc.
xeD
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/
xeD
xeD
^xeD
nho nha't cua ham so, cung nhU' cac tinh chat vc luy thiTa cua mot ba't ding
Trong thuTc te, ngtfdi ta rat hay su" dung mot triTdng hdp rieng cua tinh chat 9
Thay (4) vao (3) ta c6 maxh(x) < max l(x) - ming(x). Vay (1) dung.
xeD
xeD
Chtfng minh tinh chat nay suy tri/c tiep iii djnh nghia cua gia tri Idn nha't va
(3)
Theo tinh chat 2, ta CO: max(-g(x)) = - m i n l - ( - g ( x ) ) l = - m i n g ( x ) , (4)
xeD
\ x e D
xeD
nhiTsau:
Neu f(x) > 0 V X e D, ihi max f(x) = /minf^(x); min f(x) = /minf^(x) .
Van theo tinh chat 5 thi dau bang trong (3) xay ra khi va chi khi ton tai x,, e
xeD
D s a o c h o t a c o : maxf(x) = f(x„); max(-g(x)) =-gCx,,).
.
yueD
xeD
^xeD
Dieu nay ra't c6 ich de giai cac bai toan thuoc dang. Tim gia tri Idn nhat,
Nhi/ng max(-g(x)) = - g ( x „ ) o - ming(x) = - g ( x „ ) <=> ming(x) = gCx,,)
xeD
xeD
nho nha't cua cac ham so f(x) khi chiing diTdc cho di/di dang can bac hai hoac
xeD
=> do la dpcm.
'
CO chtfa cdc bieu thiJc vdi da'u gia tri tuyet do'i.
'
Xet thi du minh hoa sau day:
TiTdng tur ta c6 tinh cha't sau (vdi each chu'ng minh hoan toan tu'dng tiT)
' ' '
Tim gia tri Idn nha't va nho nha't cua ham so':
Ta
Gia sijf f(x), g(x) la cac ham so xac dinh va di/dng khi x e D . Dat h(x) =
•
ro
j ,
/g
maxlXx)
maxh(x)<^^^^
,
xeD
ming(x)
i
(1)
om
.
g(x) tren D. Khi do ta co:
up
va gia thict ton tai cac gia tri Idn nha't va nho nha't cua cac ham s6'h(x), f(x),
''
ce
fa
w.
maxf(x) = f(x„); ming(x) = g(x„)
xeD
V xeD
xeD
• i. u
•
•
(1),
VxeD
Ta CO f^(x) = 2 + (sinx + cosx) + 2 ^1 + (sin x + cos x) + sin x cos x .
Dat t = sinx + cosx => sinxcosx =
^ va c6 - V 2 < t < ^/2 .
2
•
= 2 + t + V2 It + 1 , vdi - ^/2 < t < >/2 *
f,:.v ^
^ Da'u b^ng trong (2) xay ra khi va chi khi ton tai Xo = D, sao cho
t
minf(x) = f(x,)); max g(x) = g(x„)
xeD
xeD
F(t)
xeD
1
P
(l->/2)t + 2 - V 2
"
1. Gia siuf f(x) la ham so xac djnh tren mien D. Khi do vdi moi n nguyen
difdng, ta co
,
,
F'(t)
V2+I
1
-
F(t)
maxf(x) = 2n+i/max(r^"^'(x)); minf(x) = 2n+i/min(f^"^'(x)) .
38
xeD
V^eR
,
xeD
\xeDV
,
_ ^
(x/2+l)t + 2+V2
F'(t)
TinhchaftS:
,
,,,
ww
xeD
,.,
m a x f ( x ) - /maxf^(x); min f(x) = /min f ^ ( x ) .
Dau b^ng trong (1) xay ra khi va chi khi ton tai Xo = D, sao cho
1
,.
X e t h a m s o F ( t ) = 2t + t + 2 l + t + -^—!- = 2 + t + V 2 V t ^ + 2 t + l
-..(.v,/,VK):iv
xeD
Do f(x) > 0 V X e R , nen theo tinh chat 9 ta co
"""" ' |« ' '
'
bo
(2)
* '"
ok
.c
xeD
minf(x)
minh(x)>^^-^^^
. '
maxg(x) ,
s/
f(x) = V l + sinx + V l + cosx , x e M.
"Iv.smy'^'*
+
1
Vay m a x f ( x ) = /max F(t) = , / m a x ( ( F ( - ^ ) ; F ( > ^ )
xeR
^\\\<42
y
(
39
Cty TNHH MTV DWH Khang Vigt
Chuygn dg BDHSG Toan gia tr| I6n nh5't va gia tr| nh6 nhgt - Phan Huy Khii
= Vmax(4 -2N/2; 4 + 2N/2) = ^ 4 + 2V2 .
V
, ,
'
m i n f ( x ) = /min F ( t ) = J F ( - 1 ) = 1.
xeD
\l\l\
f(Xo)
'
./^
max- max f(x)
xeD
,
.
,
' ^ ~ ,
Ta CO f(x) > 0 V X G D va f(x„) = 0 . Tif do suy ra min f(x) = 0.
f(Xo)
2. Neu M m > 0. Khong giam tdng quat c6 the cho la M > m > 0 (neu 0 > M
ton
xeD
Ta
om
/g
xeD
ro
max f(x), min f(x). Khi do ta c6
tai
up
tren D va
max f(x)| = max < max f(x) min f(x)
xeD
(1)
.c
xeD
ok
xeD
bo
ChiTng minh: Ap dung tinh cha't (4), thi he thiJc (1) co dang tu-dng diTdng sau:
f(x) = max ( max ff(x)
max |f(x)|
( x ) ; max (-f(x))
xeD
xeD
xeD
= max max f(x) max (-f(x))
xeD
xeD
Khi
66 ta
c6: min f(x) = min (min f(x); max f(x) I .
xeD
(xeD)
xeD2
Theo gia thiet ton tai min |f(x) va
xeD|
mm f(x) = min f ( x ) . , •
w.
|f(X().)| = - f ( X o )
.
•
-
.
c6:
f(x) max (-r(x))
xeD
;
: ; • r • xeD2
" 6 02
xeD
,
(2)
hay - f ( x ) > min l ( x ) , V x G D 2 .
> max f(x)
xeD2
Mat khac, gia s\{ max f(x) = f(X(,), x„ G D2 nen ta c6 |f(X())
xeD2
, V X G D2
max f(x) (4)
xeD2
Tijf (3), (4) va theo dinh nghla gia trj nho nhat cua ham so, ta thu di/dc
X6D2^
< max (-f(x)) < max(-f(x))
.
, ^•
(5)
min (l"(x)) = max f(x)
^ xeD
V i vay ta cung c6 |f(x„ )| < max max (-f(x))
xeD
max f(x)
xeD
Tif (4), (5) va de y rang x„ la phan tuf tijy y cua D, suy ra
40
I - (4)
i
(1)
i (
Lai theo gia thiet thi ton tai max f(x), ttJc la f ( x ) < max f(x), V x e D2,
f(x)
xeD
ww
2. Ne'u 1"(X()) < 0. Luc nay lai
1max
xeD
1
xeD|
xeD]
1. Neu f(x„) > 0. Khi do la c6 |f(x,)) = f(x„) < max f(x) < max f(x) .(3)
f(x) < max
(
Chtfng minh: V i D, = {x G D I f(x) > 0) nen |f(x)| - f ( x ) , V x G D,.
Lay tuy y x„ e D, khi do xay ra hai kha nang sau:
xeD
(9)
xeD
fa
iI xeD
ce
xeD
"eD
(2)
i
xeD
trong do D, = {x G D : f(x) > 0 } ; D2 = {x G D: f(x) < 0 ) .
s/
xeD
dinh
(8)
:v.
Tinh cha't 11: Xct ham so l(x) vdi x G D va gia suf ton tai min f(x), max l"(x),
> m chiJng minh tufdng tu"). Do do 1(1) > 0. Tuf do ta c6:
.
^
mm f(x)| = m i n f ( x ) = m = min{M,m) = m i n { | M | , | m | | =>d6 la dpcm.
xac
tiTdng
TH (6), (9) va theo dinh nghla gia tri I6n nha't cua ham so, ta c6 ngay
xeD
so
U :}Ju
f(X„),X„GD.
tokn
Tir (2), (3) suy ra f(Xo)
IXx,,) = max( max f(x) max (-f(x))
•f
ham
(7)
xcD
xeD
Difa vao tinh lien tuc cua f(x) suy ra ton tai Xo G D ma f(X()) = 0.
suT f(x) la
xeD
Gia suf max f(x)
•
'
10: Gia
max Hx)
max (-f(x))
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, ,,
1. Neu M m < 0, khi do ta c6 M > 0 > m.
Tinh cha't
(6)
xeD
(TriT^ng hdp ngiTdc lai diTdc chufng minh bang mot each hoan
[min{|M|,|m|}, neu Mm > 0.
ChiJng minh:
max (-f(x)) I .
Khong giam tdng quat c6 the cho la:
tifdng (jTng la gia trj Idn nhat, nho nhat cua ham so' l"(x) tren mien D, thi:
„
xeD
i^>:jiU
Tinh cha't 9: Gia suT l(x) la ham so xac djnh va lien tuc tren D. Khi do neu goi M , m
[O.neu Mm < 0
max f(x)
< max
(5)
xeD2
Ap dung nguycn l i phan ra (tinh chat 4) ket hdp vdi (1), (5) ta c6
max f(x)
min f(x) = mm min f(x); min f(x) 1 = min (min f(x); xeD2
xeD
• dpcm.
xeDi
xeD2
)
("eDi
ChuySn ai BDHSG Join g\& trj I6n nha't
aAut^^2.
Cty TNHH MTV DWH Khang Vi^t
gia trj nh6 nhS't - Phan Huy Khai
PHIiaNGPHllPSADMNG
+ l ) + (y + l ) + ( z + I ) ]
BifrflKiiaT H O C of T I M GUi T R I I H N N H J T T
lANHiNHAtcOAHAMSdf
(4)
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Mat khac do da'u b^ng trong (3) (ttfc la trong (2) xay ra
xac dinh gia tri Idn nha't (hoac gia trj nho nha't) cua ham so f(x). Sau do ta
.
^
o x =y= z=-.
[x + y + z = l
,,3
con CO nhiem vu la chi ra da'u bang xay ra nc'u nhiTchon du'dc Xo e D de c6
(fJ .11
(3)
Ttf (1). (3) suy r a P < - .
dang thijrc f(x) < M hoac f(x) > m V x e D, d day D la mien ma tren do ta can
?
z+ l j
y
nhaft cua ham so. De lam difdc dieu nay ta can tim cac gia tri M , m de c6 bat
^ » K
y+ 1
1
1
p„x + y + z=l.nentir(2)suyra—
—++ — +
1>I
PhiTcJng phap nay suT dung triTc tiep djnh nghia cua gia tri Idn nha't va nho
f(x„) = M hoac f^x„) = m.
(2)
>9.
x+1
iIrl:
Vay da'u b^ng trong (4) xay ra o x = y = z = - .
V i CO ra't nhieu phiTdng phap de chrfng minh ba't dang ihufc (nhiT suT dung bai
dang thifc Cosi, bat dang thuTc Bunhiacopski, phi/dng phap xuat phat tijr cac
Tir d6 ta c6 maxP =' - o x = y = z = - .
4
3
ba't dang thtfc da bie't, phu'dng phap nhom hoac them bdt cac so hang...).
Trong chufdng nay chung ta se Ian liTdt xet cac phiTdng phap cd ban nha't silf
Bai2:Chox>0,y>0va ^ +~ +
s/
Tim gia tri Idn nha't cua bieu thtfc P = —
up
ro
1.1. SiJ dung bat dang thtfc Cosi cd ban
1
Ta
dung bat dang thiJc de giai bai toan dat ra.
§1. PHaONG PHAP SUf DMNG BAT DANG THLfC COSI
j , >,
1
+V^^z
^
Hudng ddn gidi
Ap dung ba't dang thuTc Cosi cd ban hai Ian lien tiep, ta c6
ban:
1
. < i — + — . 1 — + - —+ —
2x + y + z 4 2x y + z j 4 2x"^4
/g
Ta goi hai bat dang thuTc thong dyng sau day la cac bat dang iMc Cosi cd
om
'•'
ok
.c
n > 4 vdi moi a > 0, b > 0. Da'u bang xay ra o a = b.
(a + b) fi
- +—
a b
1 1 n
(a + b + c)
> 9 vdi mpi a > 0, b > 0, c > 0.
-+—+iHffl
U
b
cj
Da'u bang xay ra o a = b = c.
'JT
ce
bo
•J'
1
1
1 ^1
- +— + —
<
—
2x + y + z 8 X 2y
2z)
1
w.
fa
Da'u hlng trong (1) xay ra o
suT dung hai bat dang thuTc noi tren.
Bai 1: Cho x > 0, y > 0, z > 0 va
X
ww
Ra't nhieu bai toan tim gia tri Idn nha't va nho nha't cua ham so quy ve viec
Tim gia tri Idn nha't cua bieu thiJc: P =
X +
o x = y = z.
1
•< —
2y + z 8 2x
y
y+1
2z;
.
1
1 1
.
—
+
—
+x + y + 2z 8
2x 2y z j
Da'u blng trong (2), (3) deu xay ra o x = y = z
z+1
Hudng ddn gidi
(2)
(3)
4
Viet lai P difdi dang sau:
1
1
+ 1-= 3y+1
z+1
^
r
-+ •
x+1
1
.
P= 1-+1
x+1
r''_
1
L i luan tiTdng tir, c6
+ y + z = 1.
(1)
C 9 n g t i r n g v e ' ( l ) ( 2 ) ( 3 ) v a c6 P < ^
1
1
.+
x+1
Ap dung ba't dang thtfc Cosi cd ban, ta c6:
y+1
-f.
z+1
(1)
Do i + i + i = 4 ^ P < i .
X
y z
4
4
-+— +•
2x 2y 2 z j
(4)
1
^
'
Chuygn de BDHSG To^n gi^ tr| Icin nhS't
Cty TNHH MTV DWH Khang Vigt
gJA tr| nh6 nhat - Phan Huy Kh5i
HUdng ddn gidi
Da'u bang trong (4) xay ra <=> dong thtJi c6 dau bang trong (1) (2) (3)
•\
1 1 1
'
<:>x = y = z = — ( k c l bdp v6i d i c u k i c n — + — + - = 4 ) .
4
X
y z
Vie't l a i b i c u thtfc P di/di dang:
^^
2x + y + z
Tur do suy ra maxP = l < = > x = y = z = ^ .
= 3 - ( X + y + z)
Nhdn xet: ThiTc chat bai toan l u y c n sinh D a i hoc Cao d^ng k h o i A - 2005 c6
y2
1
Tim gia t r i nho nhal ciia bicu ihi'rc P =
+——+
+x +y.
1- x 1- y X+y
1-y
+
x+y
,
>9i
•^.•(Ms:>.'
> -
2'
'i;''?:^'Oit^
;j T i r ( l ) ( 2 ) s u y r a P > | .
Chu y rkng k h i x = y = j
w.
=x+y o x =y= j
.
t h i x + y < 1 va liic do P = ^ . V a y minP = ^
•
C i d t r i nho n h a l dat du'dc k h i va chi khi x = y = - .
3
B a i 4: Cho ba so dudng x, y , z thoa man d i c u k i e n x + y + z = 3.
2x + y + z
44
^.'•'••''''•^••v>..^•^;•r'••'
X +
2y + z
-+•
ill
' i.-:.
1
1
1
^
• + y + 2z;
2x + y + z- + x + 2y + z + x
x + y + 2z
up
^
,
|^
K c l hdp v d i X + y + z = 3, suy ra P = - o X = y = z = 1.
V a y maxP = - . Gia t r i n^y dat di/dc k h i va chi k h i x = y = z = 1.
4
Bai 5: Cho x > 0 , y > O v a x + y = -
HUdng ddn gidi
-
2
1 1
V i e t l a i b i e u thuTc P di/di dang sau: P = - + - + — .
(1)
X
X 3y
1
1
4
A p dung bat dang thiJc Cosi cd ban, ta c6 - + — >
. (2)
X
3y x + 3y
D a u bang trong ( 2 ) xay ra o x = 3y.
. •
L a i theo bat d^ng lhi?c Cosi cd ban, la c6:
T i m gia i r i idn nhat ciia b i c u thuTc P -
1
+ x- + y + 2z
T i m gia t r i nho nhat cua b i e u thtfc P = - + ^
x 3y
(2)
ww
D a u bkng trong (2) xay r a < = > l - x = l - y
x + y + 2z
(3)
Ta
'
om
I - X > 0, 1 - y > 0.
bo
9
x+y
(1)
ce
1
1-y
2
fa
1-x
1
•+
+ —!^
ro
1-x
Theo baft dang ihifc Cosi cd ban, ta CO
1
1
1
[ ( l - x ) + ( l - y ) + (x + y ) }
+
+
•
1-x
1-y x + y
1
+ —
.c
D o X > 0, y > 0 va X + y < 1
x+y
—
1
/g
i - y
=
1
ok
1-x
1
s/
V i c t l a i P d i T d i dang
I
•+-
(1)
Tird6thayvao(l)vac6P<|.
HUihtg dan gidi
2
x + 2y + z
s9.
(2)
' ^'
Dau b^ng trong (2) xay ra o 2x + y + z = x + 2y + z = x + y + 2z o x = y = z.
x-
^ + _ !
l^2x + y + z
[(2x + y + z) + (x + 2y + z) + (x + y + 2z)]
B a i 3: Cho x X ) , y X ) va x + y < 1.
+ (I + y) + J
Jt, + y + 2z
Theo ba't d i n g thuTc Cosi cd ban, la c6:
(tufc la bai loan I'lm gia Irj li'ln nhal ma bie't trifdc dap so).
P = ( l + X ) +
x + 2y + z
iL
ie
uO
nT
hi
Da
iH
oc
01
/
^+
^+
^
< 1.
2x + y + z x + 2y + z
x + y + 2z
R6 rang diTdi dang bill dang thiirc, b a i loan de hdn ci cho l a c6 dinh hifdng
?
xx + yy +j z
1
1
1,
= 3 — [ ( 2 x + y + z ) + (x + 2 y + z ) + ( x + y + 2 z ) ]
-+
1^2x
+
y
+
z
x
+
2y
+z
4
dang sau: Cho x > 0, y > 0, z > 0 va — + — + - = 4 .
X
y z
Chtfng minh bat dang thiifc
x+y+ z
4
4
/
1
1
— +
=4 — +
2x' x + 3y
1^2x x + 3 y j
. 4
4
^ 16
hay — +
—>
2x
x + 3y
3x + 3y
<'
•.
16
2x + (x + 3y)
(3)
45
