Tài liệu tai lieu on thi dai hoc phan giai tich

CHƯƠNG 1: CÁC BÀI TOÁN VỀ HÀM SỐ
BÀI 1. PHƯƠNG PHÁP HÀM SỐ
I. TÍNH ĐƠN ĐIỆU, CỰC TRỊ HÀM SỐ, GIÁ TRỊ LỚN NHẤT & NHỎ NHẤT
CỦA HÀM SỐ
1.y=fxab⇔
( )
 
x x a b∀ < ∈

( ) ( )
 
f x f x<
2.y=fxab⇔
( )
 
x x a b∀ < ∈

( ) ( )
 
f x f x>
3.y=fxab⇔ƒ′x≥∀x∈abƒ′x=
∈ab
4.y=fxab⇔ƒ′x≤∀x∈abƒ′x=
∈ab
5.Cực trị hàm số: !"#
( )
k
x x f x
′
= ⇔
$%&

k
x
6. '(#)*&+!,&-!
• './y=ƒx)01#02ab3"#
( )

 
n
x x a b∈

45
[ ]
( )
( )
( )
( ) ( )
{ }


67 67    8
n
x a b
f x f x f x f a f b
∈
=
[ ]
( )
( )
( )
( ) ( )

{ }


6  6    
n
x a b
f x f x f x f a f b
∈
=
• 9y=fx2ab3:
[ ]
( ) ( )
[ ]
( ) ( )


6 8 67
x a b
x a b
f x f a f x f b
∈
∈
= =
• 9y=fx2ab3:
[ ]
( ) ( )
[ ]
( ) ( )



6 8 67
x a b
x a b
f x f b f x f a
∈
∈
= =

b

j j j
x x x
− ε + ε
i i i
x x x
− ε + ε
a
x
• !;&
( )
f x x= α + β
#0<
[ ]
8a b
(#)*&(#,&(=>
a; b
??@ AB9'@ C@ D6EFG?H9IJK9@ AB9'LMN9 GOL@ AB9'LMN9 
1.9P-QRS#:ux=vx)!<!<-
( )
y u x=

+*
( )
y v x=

2.9P-&QRS#:ux≥vx)!
Q=<!RST+*Q=

( )
y u x=
UVQW#0
<+*Q=
( )
y v x=

3.9P-&QRS#:ux≤vx)!
Q=<!RST+*Q=
( )
y u x=
UVQW%R*<+*Q=
( )
y v x=

4.9P-QRS#:ux=m)!<!
<-RXy=m+*
( )
y u x=

5.G@Lux≥m>∀x∈?⇔
( )
?

6
x
u x m
∈
≥
6.G@Lux≤m>∀x∈?⇔
( )
?
67
x
u x m
∈
≤
7.G@Lux≥mPx∈?⇔
( )
?
67
x
u x m
∈
≥
8.G@Lux≤mPx∈?⇔
( )
?
6
x
u x m
∈
≤
III. Các bài toán minh họa phương pháp hàm số

Bài 1. Y<!
( )

 Zf x mx mx= + −
a.L:mQRS#:ƒx=Px∈283
b.L:m&QRS#:ƒx≤P>∀x∈28[3
c.L:m&QRS#:ƒx≥Px∈
[ ]
8Z−
Giải: a.G$QRS#:ƒx=5
( )
( )
( )
( )
 
 
Z Z
 Z   Z

 
f x mx mx m x x g x m
x x
x
= + − = ⇔ + = ⇔ = = =
+
+ −

\ƒx=Px∈283:
[ ]
( )

[ ]
( )
8
8
6 67
x
x
g x m g x
∈
∈
≤ ≤

Z

]
m⇔ ≤ ≤
b.L∀x∈28[3:
( )

 Z f x mx mx= + − ≤
⇔
( )

 Zm x x+ ≤ ⇔
( )
[ ]

Z
 8[


g x m x
x x
= ≥ ∀ ∈
+

[ ]
( )
8[
6 
x
g x m
∈
⇔ ≥

^<
( )
( )

Z
 
g x
x
=
+ −
.#028[30_⇔
[ ]
( )
( )
8[


6 [
]
x
g x g m
∈
= = ≥
c.L+*x∈
[ ]
8Z−
:
( )

 Z f x mx mx= + − ≥
⇔
( )

 Zm x x+ ≥

\`
( )
[ ]

Z
 8Z

g x x
x x
= ∈ −
+
ab(c.de_5

f9
x =
:&QRS#:#V!
  Zm = ≥
0+gP
f9
(
]
8Zx∈
:G@L
⇔
( )
g x m≤
P
(
]
8Zx∈
(
]
( )
8Zx
Min g x m
∈
⇔ ≤

^<
( )
( )

Z

 
g x
x
=
+ −
.
(
]
8Z
0_
(
]
( ) ( )
8Z

Z
h
x
Min g x g m
∈
⇔ = = ≤
α
β
b
x
a
v(x)
u(x)
a
b

x
y = m
f9
[
)
8x∈ −
:

 x x+ <
0G@L
( )
g x m⇔ ≥
P
[
)
8x∈ −

[
)
( )
8
Max g x m
−
⇔ ≥
L
( )
( )
( )
[ ]



Z  
 8

x
g x x
x x
− +
′
= ≤ ∀ ∈ −
+

^<
( )
g x
0
[
)
( )
( )
8
 ZMax g x g m
−
= − = − ≥
Kết luận:ƒx≥Px∈
[ ]
8Z−
(
]
)


8 Z 8
h
m

⇔ ∈ −∞ − +∞


U

Bài 2. L:m&QRS#:5
Z
Z

Z x mx
x
−
− + − <
P>∀x≥
Giải:G@L
( )
Z 
Z [
  
Z   Z  mx x x m x f x x
x
x x
⇔ < − + ∀ ≥ ⇔ < − + = ∀ ≥

L

( )
h  h  
[  
[  [ 
   f x x x
x x x x x
−
 
′
= + − ≥ − = >
 ÷
 
_#
( )
f x
d
iYGL
( ) ( )
( )


Z      Z
Z
x
f x m x f x f m m
≥
⇔ > ∀ ≥ ⇔ = = > ⇔ >
Bài 3. L:m&QRS#:
( )


[    
x x
m m m
+
+ − + − >
>
x∀ ∈ ¡
Giải: \`
 
x
t = >
:
( )

[    
x x
m m m
+
+ − + − >
>
x∀ ∈ ¡

( ) ( )
( )
 
 [      [  [  m t m t m t m t t t t⇔ + − + − > ∀ > ⇔ + + > + ∀ >
( )

[ 
 

[ 
t
g t m t
t t
+
⇔ = < ∀ >
+ +
L
( )
( )



[ 

[ 
t t
g t
t t
− −
′
= <
+ +
0
( )
g t
#0
[
)
8+∞

_
#_⇔
( ) ( )

 
t
Max g t g m
≥
= = ≤
Bài 4. L:mQRS#:5
( )
 h [x x x m x x+ + = − + −
P
Giải: \jcP
 [x≤ ≤
G$@L
( )

h [
x x x
f x m
x x
+ +
⇔ = =
− + −

Chú ý:9W
( )
f x
′

#7b%&:<(#&QTQ%k=)l
Thủ thuật:\`
( ) ( )
Z

  

 
g x x x x g x x
x
′
= + + > ⇒ = + >
+

( ) ( )
 
h [  
 h  [
h x x x h x
x x
−
′
= − + − > ⇒ = − <
− −

E_#5
( )
g x >
+!d8
( )

h x
m+!._
( )


h x
>
+!d
⇒
( )
( )
( )
g x
f x
h x
=
dE_#
( )
f x m=
P
[ ]
( )
[ ]
( ) ( )
( )
[ ]
( )
8[
8[
 87  8 [  h  8m f x f x f f

 
 ⇔ ∈ = = −
 
 
Bài 5.L:m &QRS#:5
( )
Z
Z 
Z  x x m x x+ − ≤ − −
P
Giải:\jcP
x ≥
9e.+G@L+*
( )
Z
 x x+ − >
;Rn
&QRS#:
( )
( )
( )
Z
Z 
Z  f x x x x x m= + − + − ≤

\`
( ) ( )
( )
Z
Z 

Z 8  g x x x h x x x= + − = + −
L
( ) ( )
( )


 
Z o  8 Z  
  
g x x x x h x x x
x x
 
′ ′
= + > ∀ ≥ = + − + >
 ÷
−
 

^<
( )
g x >
+!d
x∀ ≥
8
( )
h x >
+!d0
( ) ( ) ( )
f x g x h x=
d

x∀ ≥
4&QRS#:
( )
f x m≤
P
( )
( )

  Z
x
f x f m
≥
⇔ = = ≤
Bài 6. L:m
( ) ( )

[ o x x x x m+ − ≤ − +
P>
[ ]
[ox∀ ∈ −
Cách 1. G@L
( ) ( ) ( )

 [ of x x x x x m⇔ = − + + + − ≤
>
[ ]
[ox∀ ∈ −
( )
( ) ( )
( )

( ) ( )
 

     
 [ o [ o
x
f x x x x
x x x x
− +
 
′
= − + + = − + = ⇔ =
 ÷
+ − + −
 
I;Q.0_#67
[ ]
( )
( )
[o
 oMax f x f m
−
= = ≤
Cách 2.\`
( ) ( )
( ) ( )
[ o
[ o h

x x

t x x
+ + −
= + − ≤ =

L
 
 [t x x= − + +
4&QRS#:#V!
[ ]
( )
[ ]
 
[ 8h [ 8 8ht t m t f t t t m t≤ − + + ∀ ∈ ⇔ = + − ≤ ∀ ∈
L5
( )
  f t t
′
= + >
⇒
( )
f t
d0
( )
[ ]
8 8hf t m t≤ ∀ ∈ ⇔
[ ]
( ) ( )
8h
7 h of t f m= = ≤
Bài 7. L:m

 
Z o ] Z x x x x m m
+ + − − + − ≤ − +
>
[ ]
Zox∀ ∈ −
Giải:
\`
Z o t x x= + + − >
⇒
( )
( ) ( )


Z o p  Z ot x x x x= + + − = + + −
⇒
( ) ( ) ( ) ( )

p p  Z o p Z o ]t x x x x≤ = + + − ≤ + + + − =
( ) ( )
( )
 

] Z Z o p 8 Z8Z 

x x x x t t
 
⇒ + − = + − = − ∈
 
ab

( ) ( ) ( ) ( )

Z8Z 
p

8   8 Z8Z  7 Z Z
 
f t t t f t t t f t f
 
 
 
′
= − + + = − < ∀ ∈ ⇒ = =
 
_
( )
 
Z8Z 
7 Z    q f t m m m m m
 
 
⇔ = ≤ − + ⇔ − − ≥ ⇔ ≤ − ≥
Bài 8. (Đề TSĐH khối A, 2007)
L:mQRS#:
[

Z    x m x x− + + = −
P"
Giải: \45
x ≥

$QRS#:
[
 
Z 
 
x x
m
x x
− −
⇔ − + =
+ +

\`
[
)
[
[


 
 
x
u
x x
−
= = − ∈
+ +

4
( )


Z g t t t m= − + =

L
( )

o  
Z
g t t t
′
= − + = ⇔ =
^<_0=


Z
m⇔ − < ≤
Bài 9. (Đề TSĐH khối B, 2007):YT#U5q*r
m >
QRS#:
( )

 ] x x m x+ − = −
)g>PQeP
Giải:\jcP5
x ≥

G$QRS#:5
( ) ( ) ( )
 o x x m x⇔ − + = −
( ) ( ) ( )

 
 o x x m x⇔ − + = −
( )
( )
( )
Z  Z 
 o Z  q 7 o Zx x x m x x x m⇔ − + − − = ⇔ = = + − =

_
( )
g x m⇔ =
>Pc<.
( )
8+∞
L;+;_5
( ) ( )
Z [  g x x x x
′
= + > ∀ >
^<
( )
g x
!
( )
g x
)01+!
( )
( )
 8 )
x

g g x
→+∞
= = +∞
0
( )
g x m=
>P∈
( )
8+∞

q;_
m∀ >
QRS#:
( )

 ] x x m x+ − = −
PQeP
fss
7f
Bài 10. (Đề TSĐH khối A, 2008)L:mQRS#:>P"QeP5
[ [
   o  ox x x x m+ + − + − =
Giải:\`
( )
[ ]
[ [
   o  o 8 8o f x x x x x x= + + − + − ∈
L5
( )
( ) ( )

( )
Z Z
[ [
    
 8o

 o
 o
f x x
x x
x x
   
′
= − + − ∈
 ÷
 ÷
−
 
−
 
\`
( )
( ) ( )
( )
( )
Z Z
[ [
   
8 o
 o

 o
, xu x v x
x x
x x
= − = − ∈
−
−
( ) ( )
( )
( ) ( )
( ) ( )
( )
  
  
  o
u x v x x
u v
u x v x x

> ∀ ∈

⇒ = =


< ∀ ∈


( )
( )
   

   o
 
f x x
f x x
f
′
 > ∀ ∈

′
⇒ < ∀ ∈


′
=

9:GGL@LPQeP
⇔
[
 o  o Z  om+ ≤ < +
Bài 11. (Đề TSĐH khối D, 2007):
L:mPQRS#:P
Z Z
Z Z
 
h
 
h 
x y
x y
x y m

x y

+ + + =



+ + + = −



Giải:\`
 
8u x v y
x y
= + = +

(
)
(
)
Z
Z
Z
   
Z Zx x x x u u
x x x
x
+ = + − × + = −
+!
    

  8    u x x x v y y
x x x y y
= + = + ≥ = = + ≥ =
4P#V!
( )
Z Z
h
h
]
Z h 
u v
u v
uv m
u v u v m
+ =

+ =


⇔
 
= −
+ − + = −



⇔
u v
)!P-QRS#:;
( )


h ]f t t t m= − + =
PP
( )
f t m⇔ =
P
 
t t
,t
 
8 t t≥ ≥

xofsf(x)
[
 o  o
+
I;QG.0-!
( )
f t
+*
t ≥

−∞
s  h f
∞
( )
f t
′
– –


+
( )
f t
f
∞


u[
f
∞
9:.0PP
u
  
[
m⇔ ≤ ≤ ∨ ≥
Bài 12.(Đề 1I.2 Bộ đề TSĐH 1987-2001):
L:x&QRS#:
( )

  <  x x y y+ + + ≥
>+*
y∀ ∈ ¡

Giải: \`
 <  u y y
 
= + ∈ −
 

G@L

( ) ( )
( )
( )

 
     6 
u
g u x u x u g u
 
∈ −
 
 
⇔ = + + ≥ ∀ ∈ − ⇔ ≥
 
^<
( )
y g u=
)!<X+*
 u
 
∈ −
 
0
( )
 
6 
u
g u
 
∈ −

 
≥

( )
( )


       
     
 
g x x x
x x x
g

 
− ≥ − + ≥ ≥ +
 
⇔ ⇔ ⇔

 
+ + ≥ ≤ −

≥


 

Bài 13.Y<
  
Z

a b c
a b c
≥


+ + =

YT#U5
  
[a b c abc+ + + ≥
Giải:G\L
( ) ( ) ( )
 
 
 [ Z  [a b c bc abc a a a bc⇔ + + − + ≥ ⇔ + − + − ≥
( ) ( )

  o h f u a u a a⇔ = − + − + ≥
#<
(
)
( )



 Z
 [
b c
u bc a
+

≤ = ≤ = −

9R
( )
y f u=
)!<X+*
( )


8 Z
[
u a
 
∈ −
 
 
L
( )
(
)
( )
(
)
( ) ( )

 

Z
  
  o h  8 Z   

  [ [
f a a a f a a a= − + = − + ≥ − = − + ≥

0_#
( )
8f u ≥
( )


8 Z
[
u a
 
∀ ∈ −
 
 

q;_
  
[a b c abc+ + + ≥
\XT7._#
a b c⇔ = = =

Bài 14. (IMO 25 – Tiệp Khắc 1984): 
Y<
  

a b c
a b c
≥



+ + =

YT#U5
u

u
ab bc ca abc+ + − ≤

Giải:
( ) ( ) ( ) ( ) ( ) ( ) ( )
       a b c a bc a a a bc a a a u f u+ + − = − + − = − + − =

\
( ) ( ) ( )
  y f u a u a a= = − + −
+*
(
)
( )




 [
a
b c
u bc
−

+
≤ = ≤ =
)!<X+*(
#=>
( ) ( )
( )


u

 
 [ u
a a
f a a
 + −
= − ≤ = <
 
 
+!
( )
(
)
( )
(
)
(
)


Z 

u u
    
   
[ [ u [ Z Z u
f a a a a a− = − + + = − + − ≤

^<
( )
y f u=
)!<X+*
( )


8 
[
u a
 
∈ −
 
 
+!
( )
u

u
f <
8
( )
(
)


u


[ u
f a− ≤
0
( )
u
u
f u ≤
\XT7._#

Z
a b c⇔ = = =

Bài 15. YT#U5
( ) ( )
 [a b c ab bc ca+ + − + + ≤ ∀
[ ]
   a b c∈

Giải: G$&XT+j!;&ab, c 
( ) ( ) ( )
[ ]
  [   f a b c a b c bc a b c= − − + + − ≤ ∀ ∈
\
( )
y f a=
)!<X+*

[ ]
a∈
0
( ) ( )
( )
{ }
67  8 f a f f≤
L
( ) ( ) ( )
( )
( )
[ ]
 [   [8  [ [ [    f b c f bc f a a b c= − − − ≤ = − ≤ ⇒ ≤ ∀ ∈
Bài 16. Y6M5
( ) ( ) ( ) ( )
[ ]
        a b c d a b c d a b c d− − − − + + + + ≥ ∀ ∈
Giải: G%k&XT+j!;&a, b, c, d, 5
( ) ( ) ( ) ( )
[ ]
( ) ( ) ( )
[ ]
           f a b c d a b c d b c d a b c d
= − − − − + − − − + + + ≥ ∀ ∈
\
( )
[ ]
 y f a a= ∀ ∈
)!<X0
[ ]

( ) ( )
( )
{ }

6 6   
a
f a f f
∈
=
L
( )
[ ]
     f b c d b c d= + + + ≥ ∀ ∈
( ) ( ) ( ) ( ) ( ) ( ) ( )
[ ]
( ) ( )
        f b c d b c d g b c d b c d c d
= − − − + + + ⇔ = − − − + − − + +
\
( )
[ ]
 y g b b= ∀ ∈
)!<X0
[ ]
( ) ( )
( )
{ }

6   
b

g b Min g g
∈
=
L
( )
( ) ( ) ( )
  8     g c d g c d c d cd
= + + ≥ = − − + + = + ≥
⇒
( ) ( )
[ ]
  f g b b= ≥ ∀ ∈
q;_
( )
f a ≥
_Q
BÀI 2. TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ
A. TÓM TẮT LÝ THUYẾT.
1.y=fxab⇔ƒ′x≥∀x∈abƒ′x=
∈ab
2.y=fxab⇔ƒ′x≤∀x∈abƒ′x=
∈ab
Chú ý: L#<RS#:Q$gc/%11. 2.<(!v_w,
jcPƒ′x=∈ab
CÁC BÀI TẬP MẪU MINH HỌA
Bài 1. L:m
( ) ( )

o h   Z


mx m x m
y
x
+ + − −
=
+
#02+∞
Giải: !#02+∞⇔
( )


 u
 

mx mx
y x
x
+ +
′
= ≤ ∀ ≥
+

⇔
( )
 
 u   u mx mx m x x x
+ + ≤ ⇔ + ≤ − ∀ ≥
⇔
( )


u


u x m x
x x
−
= ≥ ∀ ≥
+

( )

6
x
u x m
≥
⇔ ≥
L5
( )
( )
 
u  
 
  
x
u x x
x x
+
′
= > ∀ ≥
+


⇒ux#02+∞⇒
( )
( )

u
6 
Z
x
m u x u
≥
−
≤ = =
Bài 2. L:m
( ) ( )
Z 

 Z [
Z
y x m x m x
−
= + − + + −
#0Z
Giải. !d#0Z⇔
( ) ( )
( )

  Z  Zy x m x m x
′
= − + − + + ≥ ∀ ∈


^<
( )
y x
′
)01x=+!x=Z0⇔y′≥∀x∈2Z3
⇔
( )
[ ]

   Z Zm x x x x+ ≥ + − ∀ ∈
⇔
( )
[ ]

 Z
Z
 
x x
g x m x
x
+ −
= ≤ ∀ ∈
+

[ ]
( )
Z
67
x

g x m
∈
⇔ ≤
L5
( )
( )
[ ]


  ]
 Z
 
x x
g x x
x
+ +
′
= > ∀ ∈
+
⇒gx#02Z3⇒
[ ]
( ) ( )
Z

67 Z
u
x
m g x g
∈
≥ = =

Bài 3. L:m
( ) ( )
Z 

 Z 
Z Z
m
y x m x m x= − − + − +
#0
[
)
+∞
Giải: !d
[
)
+∞
⇔
( ) ( )

  Z   y mx m x m x
′
= − − + − ≥ ∀ ≥

⇔
( )

   o m x x x
 
− + ≥ − + ∀ ≥
 

⇔
( )
( )

 o

 
x
g x m x
x
− +
= ≤ ∀ ≥
− +

L5
( )
( )

 
 o Z

  Z
x x
g x
x x
− +
′
= =
− +



Z o
Z o
x x
x x

= = −
⇔

= = +


8
( )
) 
x
g x
→∞
=

LxGGL⇒
( )
( )


67 
Z
x
g x g m
≥

= = ≤

Bài 4.
( )
( ) ( )
Z  
 u u    Zy x mx m m x m m= − − − + + − −

[
)
+∞
Giải: !d#0
[
)
+∞
( )
 
Z   u u  y x mx m m x
′
⇔ = − − − + ≥ ∀ ≥
L
( )

u Z Zm m
′
= − +V
(
)

Z Z

u 
 [
m
 
= − + >
 
 
0
y
′
=
P
 
x x<
G@Lgx≥SjP')!5
L
( )
y x
′
≥
>
x∀ ≥
⇔
[
)
 G+∞ ⊂

( )
( )


 

h

h

 Z  Z  Z h  

o

 Z
m
x x y m m m
S m
m
′
∆ >


− ≤ ≤


′
⇔ < ≤ ⇔ = − + + ≥ ⇔ ⇔ − ≤ ≤




<
= <



Bài 5. L:m
( )

  x m x m
y
x m
+ − + +
=
−
#0
( )
 +∞
Giải: !#0
( )
 +∞
⇔
( )
 

 [  
 
x mx m m
y x
x m
− + − −
′
= ≥ ∀ >
−

⇔
( )
( )
 
 
 [    


g x x
g x x mx m m x
m
x m


≥ ∀ >
= − + − − ≥ ∀ >
 
⇔
 
≤
− ≠
 


Cách 1:Phương pháp tam thức bậc 2
L5
( )

  m
′

∆ = + ≥
_#gx=P
 
x x≤

G@Lgx≥SjP')!5
Lgx≥>∀x∈+∞⇔
( )
 G+∞ ⊂

( )
( )

 

 
    o   Z  
Z  
Z  
 

m
m
x x g m m m
m
S
m
′
≤
≤ ∆ ≥






⇔ ≤ ≤ ⇔ = − + ≥ ⇔ ⇔ ≤ −
≤ −





≥ +
= − ≤



Cách 2:Phương pháp hàm số
L5g′x=[x−m≥[x−m∀xm⇒gx#02+∞

x

x

x

x
xyfYL
^<
( )

( )
( )


 o  
Z  
6 
 Z  
Z  



x
g m m
m
g x
m
m
m
m
m
≥



= − + ≥
≤ −

≥
  


⇔ ⇔ ⇔ ≤ −
⇔
  
≥ +

≤
  

≤
≤


Bài 6. L:m
( ) ( )

[ h <  Z Z y m x m x m m= − + − + − +
.
x∀ ∈ ¡
Giải:i0=!<(
( )
h [   Z y m x m x
′
⇔ = − + − ≤ ∀ ∈ ¡
( ) ( )
[ ]
h [  Z  8g u m u m u⇔ = − + − ≤ ∀ ∈ −
^<
( )
[ ]

 8y g u u= ∈ −
)!<X0_
( )
( )
 o ] 
[

Z
   
g m
m
g m
 − = − ≤

⇔ ⇔ ≤ ≤

= − + ≤


Bài 7. L:m!
 
    Z
[ p
y mx x x x= + + +
d+*r
x∈ ¡
Giải: i0=!<(
 
< <  <Z 
 Z

y m x x x x
′
⇔ = + + + ≥ ∀ ∈ ¡
⇔
( ) ( )
 Z
 
< <  [< Z< 
 Z
m x x x x x+ + − + − ≥ ∀ ∈ ¡
( )
[ ]
Z 
[ 
 
Z 
m u u g u u⇔ ≥ − − + = ∀ ∈ −
+*
[ ]
< u x= ∈ −
L
( ) ( )


[      8 

g u u u u u u u
′
= − − = − + = ⇔ = − =
I;QGGL_#_0=!<(⇔

[ ]
( )
( )

h
67 
o
x
g u g m
∈ −
= − = ≤

Bài 8. Y<!
( ) ( ) ( )
Z 

   Z 
Z
y m x m x m x m= + + − − + +

L:mc<.-!%!U[
Giải. ab
( ) ( ) ( )

    Z  y m x m x m
′
= + + − − + =
^<

u Z m m

′
∆ = + + >
0
y
′
=
P
 
x x<
4<.-!%!U[
[ ]
   
8 8 8 [y x x x x x
′
⇔ ≤ ∀ ∈ − =

 m⇔ + >
+!
 
[x x− =
L
 
[x x− = ⇔
( ) ( )
( )
( )
( )

 
     


[   [ Z 
o [


m m
x x x x x x
m
m
− +
= − = + − = +
+
+
( ) ( ) ( ) ( )
 
[    Z  m m m m⇔ + = − + + +

u o
Z u  
o
m m m
±
⇔ − − = ⇔ =
cnQ+*
 m + >
_#
u o
o
m
+

=
B. ỨNG DỤNG TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ
I. DẠNG 1: ỨNG DỤNG TRONG PT, BPT, HỆ PT, HỆ BPT
Bài 1. '.QRS#:5
h Z
 Z [ x x x+ − − + =

Giải. \jcP5

Z
x ≤
\`
( )
h Z
 Z [ f x x x x= + − − + =

L5
( )
[ 
Z
h Z 
  Z
f x x x
x
′
= + + >
−
⇒fx#0
(



Z

−∞



6`c(f −=0QRS#:fx=P%_&x=−
Bài 2. '.QRS#:5
 
h Z  ]x x x+ = − + +

Giải. G&QRS#:⇔
( )
 
Z  ] hf x x x x= − + + − +
=
f9

Z
x ≤
:fxz⇒+gP
f9

Z
x >
:
( )
 
  

Z 
Z
] h
f x x x
x x
 
′
= + − > ∀ >
 ÷
+ +
 
⇒fx#0
(
)


Z
+∞
!f =0>Px=
Bài 3. '.&QRS#:5
Z h
[
 h u u h Z u ]x x x x+ + − + − + − <
{
Giải. \jcP
h
u
x ≥
\`
( )

Z h
[
 h u u h Z uf x x x x x= + + − + − + −
L5
( )
( ) ( )
 Z [
h
Z
[
h u Z


 
h Z u
Z h u [ u h
f x
x
x
x x
′
= + + + >
+
× −
× − × −
⇒fx#0
)
h

u


+∞


6!fZ=]0{⇔fxzfZ⇔xzZ
q;_P-&QRS#:t<)!
h
Z
u
x≤ <
Bài 4. '.@L5
Z 
  
h [ Z   h u u
 Z o
x x x x
x x x
x x x+ + + = + + − + − +
{
Giải. {
( )
(
)
( ) ( )
( )
Z 
  
h [ Z   h u u
 Z o
x x x

x x x x
f x x x x g x
⇔ = + + + − − − = − + − + =
Lfx+!g′x=−ox

+x−uz∀x⇒gx
9P-fx=gx)!<!<-
( ) ( )
+!y f x y g x= =

^<fxd8gx.+!
( ) ( )
  Zf g= =
0{P%_&x=
Bài 5. L:m67
( )
 <     < m x x x x x x
+ + ≤ + + + ∀
{
Giải. \`
( )


 <   <   t x x t x x x= + ≥ ⇒ = + = +
⇒

 t≤ ≤
⇒
 t≤ ≤
c{

⇔
( )

   m t t t t
 
+ ≤ + + ∀ ∈
 

⇔
( )


 

t t
f t m t
t
+ +
 
= ≥ ∀ ∈
 
+
⇔
( )
 
6
t
f t m
 
∈

 
≥
^<
( )
( )





t t
f t
t
+
′
= >
+

0ft
 
 
 
⇒
( )
( )
 
Z
6 

t

f t f
 
∈
 
= =
⇒
Z

m ≤
⇒
Z
67

m =
Bài 6. '.QRS#:
 
 <
] ] < 
x x
x− =
   
 <     < 
] ] <  ]  ] <
x x x x
x x x x− = − ⇔ + = +
{
ab
( )
]
u

f u u= +
L
( )
] )  
u
f u u
′
= + >
E_#
( )
f u
{
( ) ( )
   
 <  < <  f x f x x x x⇔ = ⇔ = ⇔ =


[ 
k
x k
π π
⇔ = + ∈ ¢
Bài 7. L:
( )
 x y∈ π
,tP
<  < 
Z h 
x y x y
x y

− = −


+ = π

Giải. 
<  <  <  < x y x y x x y y
− = − ⇔ − = −

ab!`#R
( )
( )
<   f u u u u= − ∈ π
L
( )


 

f u
u
′
= + >

E_#
( )
f u
#0
( )
π

4
( )
( )
[
Z h 
f x f y
x y
x y
 =
π
⇔ = =

+ = π

Bài 8. '.PQRS#:
Z 
Z 
Z 
 
 
 
x y y y
y z z z
z x x x

+ = + +

+ = + +



+ = + +

{
Giải. ab
( )
Z 
f t t t t= + +
+*
t ∈ ¡
⇒
( ) ( )


  f t t t
′
= + + >
⇒ftd
4g&W$v(./x≤y≤z
⇒
( )
( )
( )
f x f y f z≤ ≤
⇒
     z x y z x y+ ≤ + ≤ + ⇔ ≤ ≤
⇒x=y=z=±
Bài 9. '.P&QRS#:

Z
Z   

Z  
x x
x x

+ − <


− + >


Giải.


Z    
Z
x x x+ − < ⇔ − < <
\`
( )
Z
Z f x x x= − +
L5
( ) ( ) ( )
Z   f x x x
′
= − + <
⇒
( )
f x
.+!
( )

(
)
(
)
  
 
Z u Z
f x f x> = > ∀ ∈ −
II. DẠNG 2: ỨNG DỤNG TRONG CHỨNG MINH BẤT ĐẲNG THỨC
Bài 1.YT#U5
Z Z h

Z| Z| h|
x x x
x x x− < < − +
∀xm
Giải 
Z

Z|
x
x x− <
∀xm⇔
( )
Z
 
Z|
x
f x x x= − + >
∀xm

L
( )

 <
|
x
f x x
′
= − +
⇒
( )
f x x x
′′
= −
⇒
( )
 < f x x
′′′
= − ≥
∀xm
⇒
( )
f x
′′
2f∞⇒
( ) ( )
 f x f
′′ ′′
> =
∀xm

⇒
( )
f x
′
2f∞⇒
( ) ( )
f x f
′ ′
>
}∀xm
⇒
( )
f x
2f∞⇒fxmf}∀xm⇒Q

Z h

Z| h|
x x
x x< − +
∀xm⇔gx}
h Z
 
h| Z|
x x
x x− + − >
∀xm
Lg′x}
[ 
 <

[| |
x x
x− + −
⇒g′′x}
Z

Z|
x
x x− +
}fxm∀xm
⇒g′x2f∞⇒g′xmg′}∀xm
⇒gx2f∞⇒gxmg}∀xm⇒Q
Bài 2.YT#U5

 

x
x x
π
 
> ∀ ∈
 ÷
π
 
Giải.
  
  
x x
x f x
x

> ⇔ = >
π π
∀x∈


π
 
 ÷
 
abT<!
 
 
< 
 
g x
x x x
f x
x x
−
′
= =
Ve_cWPgx}x<x−x
Lg′x}<x−xx−<x}−xxz∀x∈


π
 
 ÷
 


⇒gx.#0


π
 
 ÷
 
⇒gxzg}
⇒
( )

 

g x
f x
x
′
= <
∀x∈


π
 
 ÷
 
⇒f x.#0


π
 

 ÷
 

⇒
( )
(
)


f x f
π
> =
π
⇔

  

x
x x
π
 
> ∀ ∈
 ÷
π
 
Bài 3.YT#U5
 ) )
x y x y
x y
+ −

>
−
∀xmym
Giải. ^<xmym)xm)y⇔)x−)ym0$&XT
⇔

) )  ) 

x
x y yx
x y
x
x y y
y
−
−
− > × ⇔ > ×
+
+
⇔

) 

t
t
t
−
> ×
+
+*

x
t
y
=
m
⇔

  )  

t
f t t
t
−
= − × >
+
∀tmL
( )
( )
( )
( )

 
 [ 

 
t
f t
t
t t t
−

′
= − = >
+ +
∀tm
⇒ft2f∞⇒ftmf}∀tm⇒Q
Bài 4.YT#U5

) ) [
 
y x
y x y x
 
− >
 ÷
− − −
 

( )
 x y
x y

∀ ∈


≠



Giải. abc.de_5
f9ymx:⇔

( )
) ) [
 
y
x
y x
y x
− > −
− −
⇔
) [ ) [
 
y x
y x
y x
− > −
− −
f9yzx:⇔
( )
) ) [
 
y
x
y x
y x
− < −
− −
⇔
) [ ) [
 

y x
y x
y x
− < −
− −
ab!`#Rft}
) [

t
t
t
−
−
+*t∈
L
( )
( )

  
[ 
   
t
f t
t t t t
−
′
= − = >
− −
∀t∈⇒ft
⇒fymfxymx+!fyzfxyzx ⇒Q

Bài 5.YT#U5
b a
a b<
∀amb≥~
Giải. a
b
zb
a
⇔)a
b
z)b
a
⇔b)aza)b⇔
) )a b
a b
<

ab!`#Rfx}
) x
x
∀x≥~
L
 
 )  )
  
x e
f x
x x
− −
′

= ≤ =
⇒fx2~f∞
⇒fazfb⇔
) )a b
a b
<
⇔a
b
zb
a

Bài 6. (Đề TSĐH khối D, 2007)
YT#U
( ) ( )
 
   
 
b a
a b
a b
a b+ ≤ + ∀ ≥ >
Giải. G$&XT
( ) ( )
   [  [
 
   
b a
b a
a b
a b

a b a b
   
+ +
+ ≤ + ⇔ ≤
 ÷  ÷
   
( ) ( ) ( ) ( )
( ) ( )
)  [ )  [
 [  [ )  [ )  [
a b
b a b a
a b a b
a b
+ +
⇔ + ≤ + ⇔ + ≤ + ⇔ ≤

ab!`#R<+
( )
( )
)  [
x
f x
x
+
=
+*
x >
L
( )

( ) ( )
( )

[ ) [  [ )  [

 [
x x x x
x
f x
x
− + +
′
= <
+
( )
f x⇒
.#0
( )
( ) ( )
 f a f b+∞ ⇒ ≤
Bài 7. (Bất đẳng thức Nesbitt)
YT#U5
Z

a b c
b c c a a b
+ + ≥
+ + +
∀abm
Giải. 4g&W$v(./a≥b≥\`x}a⇒x≥b≥m

L⇔f x}
x b c
b c c x x b
+ +
+ + +
+*x≥b≥m
⇒
( ) ( ) ( ) ( )
   
 
  
b c b c
f x
b c b c
x c x b b c b c
′
= − − > − − =
+ +
+ + + +

⇒fx2bf∞⇒

   
b c
f x f b
b c
+
≥ =
+


\`x}b⇒x≥m7b!gx}
x c
x c
+
+
+*x≥m
⇒
( )

  
c
g x
x c
′
= >
+
∀m⇒gx2f∞⇒
Z
   

g x g c
≥ =
Z
LxZ_#
Z

a b c
b c c a a b
+ + ≥
+ + +

∀abm
Bình luận:G&XTNesbitt#d1905+!)!&XT#&$#<
c•L#0e_)!(T&XT!_#<45 (T
Gr7~c.<=_-((T#<(5“Những viên kim
cương trong bất đẳng thức Toán học”-(.%<NXB Tri thứcQ(!(3/2009
BÀI 3. GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ
A. GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ
I. TÓM TẮT LÝ THUYẾT
1. Bài toán chung:L:(#,&<`)*&-!
( )
f x
GR*5^"<(+!T
( ) ( )
8f x c f x c≥ ≤
GR*5Y€#jcP-
( )
f x c=
2. Các phương pháp thường sử dụng
@RSQ(Q5G$!$(:QRS
@RSQ(Q5LT;
@RSQ(QZ5E/%1&XT$5Côsi; Bunhiacôpski
@RSQ(Q[5E/%1<!
@RSQ(Qh5E/%1$)Rn(
@RSQ(Qo5E/%1QRSQ(Q+bS+!Pr
@RSQ(Qu5E/%1QRSQ(Q:r+!Pr
II. CÁC BÀI TẬP MẪU MINH HỌA:
Bài 1.
L:(#,&-Pxy}x

fy


−oxyf]x−]yf
Giải. G$T%R*%
Pxy}x−Zyf[

fy−

fZ≥Z
Lx_#6@xy}Z⇔
  
Z [  
y y
x y x
− = =
 
⇔
 
− + = = −
 

Bài 2.
Y<xymL:(#,&-5S}
[ 
[ 
[ [  
y y y
x x x
y x
y x y x
+ − − + +


Giải.


 
 
   
  
y y yx
x x
S
y x
y x y x
 
 
= − + − − + + + +
 ÷
 ÷
 
 

S





 
   
y y yx x

x
y x y x
y x
 
   
 
= − + − + − + + − +
 ÷
 ÷  ÷
 ÷
 
   
 
S



 

 
 
   
y y x yx
x
y x xy
y x
 
−
 
 

= − + − + − + + ≥
 ÷
 ÷
 ÷
 
 
 

q*x}ym:6E}
Bài 3.
L:(#)*&-!
  
    S x y x y
= + + +
Giải .

  
    S x y x y
= + + +
}

 <  < 
 <  
 
yx
x y
−−
+ + − +
S
 

p

 <  <  <   < <  <  
[ [
x y x y x y x y x y x y
 
= − + − − + = − + + − + +
 
 
S


p p
 
<  <    
[  [ [
x y x y x y
 
= − − + + − − ≤
 
 

q*
Z
x y k
π
= = + π k∈:
p
67
[

S = 
Bài 4. L:(#,&-T
   
  Z ]    Z o u u ] ]
   S x x x x x x x x x x x x x= + + + + − + + + + +
Giải.
   
   Z Z [ [ h
 Z  [ Z h [
 [ Z o [ ] h
S x x x x x x x x
       
= − + − + − + − +
 ÷  ÷
 ÷  ÷
   
   
   
h o o u u ] ]
o h u o ] u p ] [ [
 o  u [ ] o p p p
x x x x x x x
       
+ − + − + − + − − ≥ −
 ÷  ÷  ÷  ÷
       
q*
   Z o u u ] ]
  o u ]
8 88 8 8

 Z u ] p
x x x x x x x x x= = = = =
:
[
6
p
S = −
Bài 5.Y<
 x y z ∈ ¡
L:(#,&-T5
E}px

fh[y

foz

−oxz−[yfZoxy
Giải. G$E⇔fx}px

−]z−]yxfh[y

foz

−[y
L∆′
x
}gy}]z−]y

−h[y


foz

−[y}−uy

fo]zy−[z


⇒∆′
y
}][z

−u[z

}−o[[z

≤∀z∈M⇒gy≤∀yz∈M
E_#∆′
x
≤∀yz∈M⇒fx≥q*
x y z= = =
:
MinS =
Bài 6.Y<x

fxyfy

}ZL:(#)*&+!,&-T5
E}x

−xyfy


Giải aby}⇒x

}Z⇒E}Z)!(#-!
aby≠c$T%R*%e_
( )

 

   
    

Z
       
x y x y
x xy yS t t
u u
x xy y x y x y t t
− +
− + − +
= = = = =
+ + + + + +
+*
x
t
y
=
⇔ut

ftf}t


−tf⇔u−t

fuftfu−}{
f9u}:t}⇒x}y}
Z±
⇒u})!(#-!
f9u≠:u;Q(#!⇔QRS#:{Pt
⇔∆}Zu−Z−u≥⇔

 Z
Z
u≤ ≠ ≤

q;_;Q(#-u)!

Z
Z
 
 
 
⇒

6
Z
u =
867u}Z
6E}⇔

6

Z
u =
⇔t}⇒
 

Z
x y
x y
x xy y
=


⇔ = = ±

+ + =



67E}p⇔67u}Z⇔t}−⇒
 
Z Z
Z
Z Z
x y
x y
x xy y
x y

= −


= = −


⇔


+ + =

= − =


Bài 7.Y<xy∈M,tjcP
( ) ( )

     
  [  x y x y x y− + + − + =
 L:(#)*&,&-TE}
 
x y+
Giải. G$
( ) ( ) ( )

       
  [ x y x y x y x y− + − + + − + =
⇔
( ) ( )

    
Z  [ x y x y x+ − + + + =
⇔

( ) ( )

    
Z  [x y x y x+ − + + =−
^<−[x

≤0
( ) ( )

   
Z  x y x y
+ − + + ≤
⇔
 
Z h Z h
 
x y
− +
≤ + ≤
q*x}y}
Z h

−
±
:
 
Z h
6 

x y

−
+ =

q*x}y}
Z h

+
±
:
 
Z h
67 

x y
+
+ =
Bài 8
.
L:(#,&-!
( )

[  f x x x x= + + +

Giải.
'ry

)!(#-!fx
⇒x

<<y


}

  
[  x x x+ + +

⇔
   
         
[    [  y x x x y y x x x x− = + + ⇒ − + = + +

⇔gx

}
 
   
Z    x y x y+ + + − =
Lgx}Px


⇔∆′}
  
   
  Z   y y y y+ − − = + −
}
 
   y y+ − ≥
^<y

}

  
      
Z   Z Z x x x x x x x+ + + ≥ + = + ≥
0
∆′≥⇔y

−≥⇔



y ≥
q*x}


−
:6fx}


Bài 9.Y<
( )

h [ y f x x x mx
= = − + +
L:((#-m<<
6 y
>
Giải. L
( )
( )
( )

( )
( )




h [ 8 7  [ 5
h [ 8  [ 5
x m x x P
f x
x m x x P

+ − + ≤ ∨ ≥

=

− + + − ≤ ≤


'rP)!-y}fx⇒P}P

∪P

cP#<(:%
e_

Hoành độ của các điểm đặc biệt trong đồ thị (P):
<!<P

P


x
A
}8x
B
}[8 <!€P

5
h

C
m
x
−
=

9:+!<7b(c.d5
9x
C
∈2x
A
x
B
3⇔m∈2−ZZ3:6fx}6{ff[}
46fxm⇔
Z Z
 
[ [ 
m
f m

f m
− ≤ ≤



= >


= >


⇔zm≤Z
9x
C
∉2x
A
x
B
3⇔m∉2−ZZ3:6fx}
( )
 
h

C
m
f x f
−
 
=
 ÷

 
}

 p
[
m m
− + −
46fxm⇔

2 ZZ3
Z h  Z
 Z 
m
m
m m
∉ −


⇔ < < +

− + <



Kết luận
:
Lx+!_#6fxm⇔
325m1
+<<


Bài 10. (Đề thi TSĐH 2005 khối A)
Y<
  x y z >
8
  
[
x y z
+ + =
L:6-E
  
  x y z x y z x y z
= + +
+ + + + + +
Giải:E/%1&XTYg<(a, b, c, d > 5
( )
(
)
[
[
o
        
[ [ oa b c d abcd
a b c d abcd a b c d a b c d
+ + + + + + ≥ = ⇒ + + + ≥
+ + +
o o
   

o o
   


o o
   

     
o [ o 6 
  
x x y z x x y z x y z
x y y z x y y z x y z
x y z z x y z z x y z
S
x y z x y z x y z x y z

+ + + ≥ =

+ + + + +


+ + + + ≥ =

+ + + + +


+ + + ≥ =

+ + + + +

   
= + + ≥ + + ⇒ =
 ÷  ÷

+ + + + + +
   
•
G
Y
P

P

•
G
Y
P

P

•
G
Y
P

P

Bài 11. (Đề thi TSĐH 2007 khối B)
Y<
  x y z >
L:6-E
  
  
y

x
z
x y z
yz zx xy
 
 
 
= + + + + +
 ÷
 ÷  ÷
 
   
Giải: E/%1&XTYg<p
E
[ [ [
  
p
[ [ [
p p p

 6
   
y y x y z
x x
z z
x y z S
yz yz zx zx xy xy
x y z
 
= + + + + + + + + ≥ = ⇒ =

 ÷
 
Bài 12.
Y<
 

x y
x y
>



+ =


L:(#,&-S}
 
yx
x y
+
− −

Giải:
( ) ( ) ( )

yx
S y x x y x y x y x y
y x
 
 

= + + + − + ≥ + − + = +
 ÷ ÷
 
 
6`c(S}
 
yx
x y
+
− −
}
 y x
y x
− −
+
}
( )
 
x y
x y
 
+ − +
 ÷
 ÷
 

E_#S≥
 
x y
+

≥
[
 
 

xy x y
≥ =
+
⇒
S ≥
⇒6S}


Bài 13.
Y<xy‚mL:67-5S}
( )
( )
  
  
 
xyz x y z x y z
x y z xy yz zx
+ + + + +
+ + + +

Giải: E/%1&XTCôsi +!BunhiaCôpski Z((5
     
Z
Zx y z x y z+ + ≥ ×
8

ƒ6Gƒ^ƒv<Z
  
Z
Z
Z   Zxy yz zx xy yz zx x y z+ + ≥ =
( )
( )
        
   Zx y z x y z x y z+ + ≤ + + + + = + +
Lx_#
( )
( )
  
Z Z
         Z
Z
 Z
 Z  Z Z Z
Z Z p
Z
Z
xyz x y z xyz xyz
S
xyz
x y z x y z x y z
+ + +
+ + +
≤ = × ≤ × =
+ + + +
Bài 14. (Đề thi TSĐH 2003 khối B)

L:(#)*&,&-!

[y x x= + −
Cách 1: L;Q7(
[ ]
8D = −
8



 8  [
[
x
y y x x
x
′ ′
= − = ⇔ = −
−
 


[
x
x
x x
≥


⇔ ⇔ =


= −


⇒
7  

 
y
y

=


= −


Cách 2: \`
  8
 
x u u
π π
 
= ∈ −
 
 
⇒
( )
(
)
  <    8  

[
y u u u
π
 
= + = + ∈ −
 
8
7  8   y y= = −
Bài 15. (Đề dự bị TSĐH 2003 khối B)
L:(#)*&,&-
( )
Z
o 
[ y x x= + −
#0<
[ ]
8−
Cách 1.\`
[ ]

8u x= ∈
L
( )
Z
Z Z 
[  Z   [y u u u u u= + − = − + − +
[ ]

 


p [   8 8  
Z
y u u u u
′
= − + − = ⇔ = ∈ = >
9:.0
[
7 [8
p
y y= =
Cách 2. \`
o o
  [ <x u y u u= ⇒ = +

xy′−+
y[

x−y′f −
y
−

( ) ( )
o o o  
 < Z<  < Z [u u u u u= + + ≤ + + =
q*
x =
:
7 [y =
E/%1&XTYg5
o o 

Z
o o 
Z
] ] ] ]
[
 Z  
u u u u Z
[ [ [ [ [
[< Z [< <
u u u u Z
u u u
u u u

+ + ≥ × × × =




+ + ≥ × × × =


( )
o o  
]
[ [ [
 [ <  <
p Z Z p
y u u u u y= + + ≥ + = ⇒ ≥
q*
 [


Z p
x y= ⇒ =

Bài 16.I;Q.0+!:(#)*&-!

Z

x
y
x
+
=
+
Y<
a b c+ + =
YT#U5
  
   a b c+ + + + + ≥
Giải. La\5
D = ¡
8
( )
(
)
 
 Z
 
 
Z Z

 
x
y x y
x x
−
′
= = ⇔ = ⇒ =
+ +
( ) ( )



Z  Z 
) ) ) )



x x x x
x x x x
x
y
x
x
x
x
→∞ →∞ →∞ →∞
+ +
= = =
+
+


E_#
) 8 ) 
x x
y y
→+∞ →−∞
= = −
9:GGL


Z
 7 

x
y y
x
+
= ≤ ⇒ =
+

L~<Q=:
 y x≤ ∀
⇔

Z  x x x+ ≤ + ∀

\`P&XT!_((#
 x a x b x c= = =
5




5 Z  
5 Z  
5 Z  
x a a a
x b b b
x c c c

= + ≤ +



= + ≤ +


= + ≤ +

( )
  
p    a b c a b c
+ + + ≤ + + + + +
⇔
  
   a b c
≤ + + + + +
Cách 2. L#0`QXr„7_`
( ) ( ) ( )
8 8 8 8 8OA a AB b BC c= = =
uur uuur uuur


4
( )
8 ZOC OA AB BC a b c= + + = + +
uuur uur uuur uuur

^<
OA AB BC OA AB BC OC+ + ≥ + + =
uur uuur uuur uur uuur uuur uuur
Lx_#
  
   a b c+ + + + + ≥
Bài 17. (Đề 33 III.2, Bộ đề thi TSĐH 1987 – 1995)
Y<
 
x y+ =
L:676-A=
 x y y x+ + +

Giải. 1. L:MaxA: E/%1&XTBunhiaCôpski 
A≤
( )
( )
( )
( )
   
      x y y x x y x y
 
+ + + + = + + ≤ + + = +
 


q*


x y= =
:67A=
 +
2. L:MinA:ab#RnQe_
…Trường hợp 159
xy ≥
7bc.d5
xZ y′f −
y
−

a
a+ba+b+c
C
A
B


Z
O x

y
f9
 x y≥ ≥
:•m⇒
6 A >

f9x≤y≤:
|A|≤
[ ]
 
      x y x y x y+ + + + = + +
}
( )
 
  x y x y− − ≤ − + =
Lxc.dt7b_#+*
xy ≥
:6A}−
…Trường hợp 25ab
xy <
5\`
x y t+ =
⇒




t
xy
−
= <
⇒
( )
t ∈ −
( )
( )

( )
( )
( )
  
       A x y xy x y y x xy x y xy x y xy
= + + + + + + = + + + + + +
=
  
  
  
  
t t t
t t
− − −
+ × + × + +

( )


   

t
t
−
 
= + + +
 
⇔
( )
( ) ( )

 Z 

      

A f t t t t
 
= = + + − + + −
 
L5
( )
( )

 
Z  
   
  8  
  Z
f t t t t t t t
+
+ +
′
= + − = ⇔ = = − = = −
L
 
t t
+!<Q=%R-
( )
f t
<
( )

f t
′
⇒
( )
( )
( )
 
 p Z 
8 
u
f t f t
−
= =

9:.0_#5
( ) ( )

 
A f t A f t≤ ⇒ ≥ −
_#
( )
( )

 p Z 
6 
u
A f t
−
= − = − < −
7._#⇔


x y t+ =
8




t
xy
−
=
⇒xy)!P-

   Z

Z p
u u
+ −
+ + =
⇒
( )
  h  

o
x y
− + ± −
=
Kết luận: 67A=
 +
8

( )
 p Z 
6
u
A
−
= −
Bài 18.Y<
[ ]
  x y z ∈
<.tjcP5
Z

x y z+ + =

L:676-T5
( )
  
<S x y z= + +

Giải. ^<
[ ]
  x y z ∈
0
  
Z

 
x y z x y z
π

< + + < + + = <

q:!
<y = α
#0
(
)


π
0!<(#V!
1. L:MaxS_:Min
( )
  
x y z+ +
( )
( )
( )

        
Z

  
Z [
x y z x y z x y z+ + = + + + + ≥ + + =

q*


x y z= = =

:67E}
Z
<
[
2. L:MinS _:Max
( )
  
x y z+ +
Cách 1: Phương pháp tam thức bậc hai:
4g&W$v(./
{ }

  8

z Max x y z z
 
= ⇒ ∈
 
 
G$+!((R+j
T;‚

( )
(
)
( )


     
Z p

  Z
 [
x y z z x y xy z z z z f z+ + = + + − ≥ + − = − + =
t−t

t

ƒ′+−+ƒ
^<!_}f‚)!Q#<)v_j)†)0#005
( )
(
)
( )
{ }
(
)
( )
h
 
67 67 8  
  [
f z f f f f= = = =

q*

8 8 

z x y= = =
:6E}
h

<
[
Cách 2: Phương pháp hình học
abPr\j(+g„7_‚L;QnQ(
( )
 M x y z
<.tjcP
[ ]
  x y z ∈
U#<:);QQRS•GY^•′G′Y′„+*•8G8Y
8^8•′8G′8Y′
6`c(%<
Z

x y z+ + =
0
( )
 M x y z
U#0`QX@5
Z

x y z+ + =
q;_;QnQ(
( )
 M x y z
<.tjcP.U#0%Pƒ?‡4I9+*
(ƒ?‡4I9)!#(:);QQRS'r„′)!:-„)0
ƒ?‡4I9:„′)!e-:);QQRS+!ˆ)!e-)1(jƒ?‡4I9L„′6
)!:-„6)0ƒ?‡4I9^<„6


}
  
x y z+ +
0„6)*&⇔„′6)*&
⇔6#‰+*#<o€ƒ?‡4I9
Lx_#5
(
)
   
h


[ [
x y z OK+ + ≤ = + =
( )
(
)
  
h
< <
[
x y z⇒ + + ≥
q*

8 8 

z x y= = =
:6E}
h
<

[
Bài 19. Y<
a,b,c 0
>
,tjcP
3
a b c
2
+ + ≤
L:(#,&-
  
  
  
S a b c
b c a
= + + + + +
Giải. Sai lầm thường gặp:

     
Z
o
     
     
Z ZS a b c a b c
b c a b c a
   
≥ + × + × + = + + +
 ÷ ÷ ÷
   


o
  
o
  
  
Z    Z ] Z  6 Z a b c S
b c a
   
≥ × × × × × × = = ⇒ =
 ÷ ÷ ÷
 ÷ ÷ ÷
   
• Nguyên nhân:
   Z
6 Z   Z

S a b c a b c
a b c
= ⇔ = = = = = = ⇒ + + = >
el+*.
• Phân tích và tìm tòi lời giải :
^<E)!T7T+*abc0%"<(6E


a b c
= = =

Sơ đồ điểm rơi:
y
Z

„
ƒ


4
Z
‡
6
z
x
?
I
9
Z

„′



a b c
= = =
⇒
  
  

[
   [
a b c
a b c


= = =




= = =

α
α α α

⇒
 [
[
=
α
⇒
16
α =
Cách 1:G$+!/%1&XTCôsi ta có
  
     
o o o
     
  
o o o o o o
S a b c
b b c c a a
= + + + + + + + + + + +
1 4 4 2 4 4 3 1 4 4 2 4 4 3 1 4 44 2 4 4 43
  

u u u
u u u
o Z o Z o Z ] o ] o ] o
u u u u
o o o o o o
a b c a b c
b c a b c a
 
≥ × + × + × = + +
 
 
Z
u u u u
] o ] o ] o ] h h h

u Z Z u
o o o o
a b c
b c a a b c
 
 
≥ × × × =
 
 
(
)
u
h h
u
Z u Z u Z u


    
  

Z
a b c
a b c
= ≥ ≥
×
+ +
×
q*


a b c
= = =
:
Z u
6

S
=
Cách 2:G$+!/%1&XTBunhiaCôpski 

( )
( )
( )
   
 
   

 
   
 
    [
 [
u u
    [
 [
u u
    [
 [
u u
a a a
b
b b
b b b
c
c c
c c c
a
a a

   
+ = × + + ≥ × +

 ÷  ÷
   




   
+ + = × + + ≥ × +

 ÷  ÷
   


   

+ = × + + ≥ × +
 ÷  ÷

   

⇒
 [ [ [
u
S a b c
a b c
 
≥ × + + + + +
 ÷
 
    h   
[ [ [ [
u
a b c
a b c a b c
 
 

= × + + + + + + + +
 ÷
 
 
 
o Z
Z
    h     [h 
o Z Z
[ [ [ [ [
u u
abc
a b c a b c
abc
 
 
 
≥ × × × × + × × × = + ×
 
 ÷
 ÷
 
 
 
 [h   [h Z u
Z Z 
[ [ 
u u
Z
a b c

   
≥ + × ≥ + × =
 ÷
 ÷
+ +
 
 ÷
 
q*


a b c
= = =
:
Z u
6

S
=
Cách 3:\`
(
)
(
)
(
)
  
   8    8   u a v b w c
b c a
= = =

uur uur uuur
^<
           u v w u v w+ + ≥ + +
uur uur uuur uur uur uuur
0_#5
( )


  
  
     
S a b c a b c
a b c
b c a
 
= + + + + + ≥ + + + + +
 ÷
 
=
( )
 

    h   
o o
a b c
a b c a b c
   
+ + + + + + + +
 ÷  ÷
   


≥
( )
(
)

Z
h
      
 Z
[
o
a b c
a b c a b c
 
+ + × × + + + × × ×
 ÷
 

≥
( )
Z
Z

Z
 Zh 
  
Z Z
 o
abc

a b c
abc
× × × × × × + ×
≥
(
)

p Zh

 o
Z
a b c
+ ×
+ +
≥
p Zh ] Zh hZ Z u
[
 o [ [ [ 
+ × = + = =
q*


a b c= = =
:
Z u
6

S =
B. CÁC ỨNG DỤNG GTLN, GTNN CỦA HÀM SỐ
I. ỨNG DỤNG TRONG PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH

Bài 1. '.QRS#:5
[ [
 [ x x− + − =
Giải. \`
( )
[ [
 [f x x x= − + −
+*
 [x≤ ≤

( )
( ) ( )
Z Z
[ [
  
 Z
[
 [
f x x
x x
 
′
= − = ⇔ =
 
− −
 
9:GGL_#5
( ) ( )
[ ]
Z  [f x f x≥ = ∀ ∈

⇒@RS#:
( )
[ [
 [ f x x x= − + − =
P%_&x=Z
Bài 2. '.QRS#:5
Z h o 
x x
x+ = +

Giải. @L ⇔
( )
Z h o  
x x
f x x= + − − =
L5
( )
Z ) Z h ) h o
x x
f x
′
= + −
⇒
( ) ( ) ( )
 
Z ) Z h ) h 
x x
f x
′′
= + >


x∀ ∈ ¡
⇒ƒ′x
6`c(ƒ′x)01+!
( )
 ) Z ) h o f
′
= + − <

( )
 Z) Z h) h o f
′
= + − >
⇒@RS#:ƒ′x=>Px


9:.0_#5
@RS#:
( )
Z h o  
x x
f x x= + − − =
cgv(P
6!
( )
( )
  f f= =
0QRS#:>P
x =
+!

x =
Bài 3. L:mG@L5

 pm x x m+ < +
P>
x∀ ∈ ¡
Giải.

 pm x x m+ < +
⇔
( )

 p m x x+ − <
⇔
( )

 p 
x
m f x
x
< =
+ −
L5
( )
( )


 
p  p
 p  p 

x
f x
x x
− +
′
=
+ + −
=⇔

 p p ox x+ = ⇔ = ±
( )

 
) )
p 


x x
f x
x
x
→+∞ →+∞
= =
+ −
8
( )

 
) )
p 



x x
f x
x
x
→−∞ →−∞
− −
= =
+ +
9:GGL
( )
f x m>

x∀ ∈ ¡

⇔

( ) ( )
Z Z
6 o
[ [
x
f x f m m
∈
−
= − = − > ⇔ <
¡
x−∞x


+∞f′−+
f
ƒx


x−∞−oo+∞f′−f−
ƒ
xZ[ƒ′−+
ƒ
Bài 4. L:m@L5
( )

    <x m x+ = +
P

 
x
π π
 
∈ −
 
 
Giải. ^<

 
x
π π
 
∈ −
 

 
⇒

 [ [
x −π π
 
∈
 
 
0`
[ ]
 

x
t = ∈ −
⇒



<

t
x
t
−
=
+
8





t
x
t
=
+
4⇔
( ) ( )
 
  <  <x x m x+ = +
⇔
( )
( )
 
 


 
  
    
 
t t t
m f t t t m
t t
   
+ − −
= + ⇔ = + − =
 ÷  ÷
+ +

   

L5
( )
( )
( )

      8  f t t t t t t
′
= + − − = ⇔ = = −
⇒G.0
9:.0_#5
\P
[ ]
t ∈ −

:
[ ]
( )
[ ]
( )


6  67
t
t
f t m f t
∈ −
∈ −
≤ ≤


⇔
  [  m m≤ ≤ ⇔ ≤ ≤
q;_P

 
x
π π
 
∈ −
 
 
:
[ ]
8 m∈

Bài 5. L:mPG@L5

Z 
Z 
  [ 
x x
x x x m m

− ≤



− − − + ≥


P
Giải. ⇔
( )
Z 
 Z
  [
x
f x x x x m m
≤ ≤



= − − ≥ −



L5
( )
[
)
(
]


Z [ [ 8 
Z [ [ 8Z
x x x
f x
x x x


+ − ∀ ∈

′
=


− + ∀ ∈

8
ƒ′x=⇔

Z
x =
9:GGL_#5
[ ]
( ) ( )
8Z
67 Z 
x
f x f
∈
= =
\P:
[ ]
( )

8Z
67 [
x
f x m m

∈
≥ −
⇔

[ m m− ≤
⇔−Z≤m≤u
Bài 6. L:m≥P5
Z 

Zh
 < o
[
ZZ
<  o
[
x y m m m
x y m m

= − − +




= − +


P
Giải
⇔
Z

Z 
 < <   u

 < <  

x y x y m m
x y x y m m

+ = − +


− = − +


⇔
( )
( )
Z
Z 
  u

 

x y m m
x y m m

+ = − +


 − = − +



ab
( )
Z
 uf m m m= − +
L5
( )

Z    f m m m
′
= − = ⇔ = >
9:GGL_#5ƒm≥ƒ=∀m≥
cnQ+*
( )
 x y+ ≤
_#P
P:m=cP#V!5
( )
( )
 



x y
x y

+ =



− =


P
8
Z o
x y
π π
= =
q;_P
⇔
m=
xZf′−++fYL]
t−ƒ′t−+
ƒt[
[
m+∞ƒ′−+
ƒu+∞
II. ỨNG DỤNG GTLN, GTNN CHỨNG MINH BẤT ĐẲNG THỨC
Bài 1. YT#U5
( )
 
 )  x x x x+ + + ≥ +

x∀ ∈ ¡
G\L⇔
( )
( )
 
 )   f x x x x x= + + + − + ≥


x∀ ∈ ¡
L5
( )
( )

)   f x x x x
′
= + + = ⇔ =

⇒G.0
9:.0_#5
( ) ( )
 f x f≥ =
⇒Q
Bài 2. Y<
  
  

a b c
a b c
>



+ + =


Y6M5T=
     

Z Z

a b c
b c c a a b
+ + ≥
+ + +
L5T=
( ) ( ) ( )
  
  
  
  
  
a b c a b c
a b c
a a b b c c
+ + = + +
− − −
− − −

ab!
( )
( )

f x x x= −
+*xm
L
( )



 Z  
Z
f x x x
′
= − = ⇔ = >

9:.0⇒
( )


Z Z
f x x≤ ∀ >

45
( ) ( ) ( )
( )
  
  
Z Z Z Z
 

a b c
T a b c
f a f f c
= + + ≥ + + =
\XT7._#

Z
a b c⇔ = = =


Bài 3. Y<Z≤n)ŠYT#U5∀x≠5
( ) ( )
  Z
    
| | | Z| |
n n
x x x x x
x x
n n
+ + + + − + − + − <
\`
( ) ( )
  Z
  8  
| | | Z| |
n n
x x x x x
u x x v x x
n n
= + + + + = − + − + −

L=T
( ) ( ) ( )
f x u x v x=
z
L5
( )
( )
( )
( )

( )
( )
 
 
 
| |
 |
 
| |
 |
n n
n n
x x x
u x x u x
n
n
x x x
v x x v x
n
n
−
−

′
= + + + + = −

−




′
= − + − + − = − −

−

⇒
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
 
| |
n n
x x
f x u x v x u x v x u x v x u x v x
n n
   
′ ′ ′
= + = − − +
   
   
⇒
( ) ( ) ( )
[ ]
|
n
x
f x u x v x
n
−
′
= +


( )
 [ 

 
| | [|
 |
n n
x x x x
n
n
−
 −
= + + + +
 
−
 
^<Z≤n)Š0ƒ′x‰%&+*−x
9:.0_#5
x−∞+∞f′−+
f

x−∞+∞f′+−
f
x−∞+∞f′+−
f
( ) ( )
  f x f x< = ∀ ≠
⇒Q
Bài 4.YT#U5
Z Z [ [

Z [
 
a b a b+ +
≤
∀abm
(
)
(
)
[
[
[ [
[ [ [
[ [
Z Z
Z Z
Z Z Z Z
Z

  
 


a
a b t
b
a b t
a
b
+

+ +
≥ ⇔ = ≥
+ +
+

abft}
( )
( )

[
[ [
[

Z
Z
Z
Z
 


t t
t
t
+ +
=
+
+
+*

a

t
b
= >
f′t}
( ) ( ) ( ) ( )
( )

Z
 
[ Z Z [  Z
[ [ Z
Z

Z
Z
   

t t t t t t
t
−
−
+ + − + +
+
( )
( )
( )
( )

Z


Z
 [
[
Z

Z
Z
  

t t t t
t
−
−
+ + −
=
+
f′t}⇔t}⇒G.0-ft
LxGGL⇒
[
Z


≤ftz∀tm⇒
[
[ [
[
Z
Z
Z Z



a b
a b
+
≤
+
⇒
Z Z [ [
Z [
 
a b a b+ +
≤ 
^&U7._#⇔a}bm
III. BÀI TẬP VỀ NHÀ
Bài 4. Y<∆•GY
A B C> >
L:(#,&-!5
( )
 

 
x A x B
f x
x C x C
− −
= + −
− −
Bài 5. L:676-5 y=
o o
 <  <x x a x x+ +


Bài 6. Y<ab≠L:6-
[ [  
[ [  
a b a b a b
y
b a
b a b a
 
= + − + + +
 ÷
 
Bài 7. Y<
 
x y+ >
L:676-
 
 
[
x y
S
x xy y
+
=
+ +
Bài 8. './QRS#:



x px

p
+ + =
Px

x


L:p≠<<
[ [
 
S x x= +
,&
Bài 9. L:6-
( ) ( ) ( ) ( )
 
 Z  Z ]  Z  Z
x x x x
y
 
= + + − − + + −
 
Bài 10. Y<xy≥+!
x y+ =
L:676- Z p
x y
S = + 
Bài 11. Y<
  
x y z+ + =
L:676-

P x y z xy yz zx= + + + + +

Bài 12. L:m@L5
( ) ( )
   x x x x m− + + − − + =
P
Bài 10 L:m@L5

p px x x x m+ − = − + +
P
Bài 11 L:m@L5
( )
Z
  
  [    [x x x x x x m− + − − + = − +
[PQeP
Bài 12 L:m@L5

Z 
 
 
x
x mx
x
−
= − +
−
P%_&
Bài 13 L:m@L5
<  [ <  m x x x m− + − =

P
(
)

[
x
π
∈

tf∞f
′
−ff
