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
′
= ⇔
$%&

{ }


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:
[ ]
( ) ( )
[ ]
( ) ( )



+*
( )
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∈?⇔
( )
?

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:
[ ]
( )

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

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

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


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

[    
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
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
′

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

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

( ) ( )
 

     
 [ 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

 
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

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

 
 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

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

   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

  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#:;
( )

+
( )
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

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

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



+ + =

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
−

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

[ ]
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
[ ]

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


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

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
 
− + ≥ − + ∀ ≥
 

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

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

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

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

 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

[ ]
( )
( )

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


[   [ 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
+


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 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

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{

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

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

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

+ = + +

+ = + +



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
.+!
( )

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

[| |
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

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

>
−
∀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
−
> ×
+
+*

′
= − = >
+ +
∀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
− < −
− −
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

= ≤ =
⇒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

( ) ( )
( )

[ ) [  [ )  [

 [
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

\`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
+ + ≥
+ + +


−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
+ − − + +

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

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
[

 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



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

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

= −


( ) ( )

    
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


}

  
[  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

}

( )
( )




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

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
−
 
=
 ÷

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
   

+ + + + + +
   
•
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

 ÷
 
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
 
 

≥
[
 
 

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

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

[
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= + − = − + − +
[ ]

 


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*
 [

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



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


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 >

( )
( )
( )
  
       A x y xy x y y x xy x y xy x y xy
= + + + + + + = + + + + + +
=
  
  
  
  
t t t
t t
− − −
+ × + × + +

( )


   

t
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._#⇔

( )
 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

: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

<
[
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


[
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
   
≥ + × + × + = + + +
 ÷ ÷ ÷
   


„
ƒ


4
Z
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z
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9
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[
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a b c
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u u u u
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a b c
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(
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
p Zh
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p Zh ] Zh hZ Z u
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a b c= = =
:
Z u
6
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B. CÁC ỨNG DỤNG GTLN, GTNN CỦA HÀM SỐ
I. ỨNG DỤNG TRONG PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH

⇒@RS#:
( )
[ [
 [ f x x x= − + − =
P%_&x=Z
Bài 2. '.QRS#:5
Z h o 
x x
x+ = +
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( )
Z h o  
x x
f x x= + − − =
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( )
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x x
f x
′
= + −
⇒
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x x
f x
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x =
Bài 3. L:mG@L5

 pm x x m+ < +
P>
x∀ ∈ ¡
Giải.

 pm x x m+ < +
⇔
( )
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 p m x x+ − <
⇔
( )

 p 
x
m f x
x
< =
+ −
L5
( )
( )


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p  p
 p  p 


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x x
f x
x
x
→−∞ →−∞
− −
= =
+ +
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( )
f x m>
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x∀ ∈ ¡
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⇔
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x
f x f m m
∈
−
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¡
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x −π π
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∈
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0`
[ ]
 
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x
t = ∈ −
⇒
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t
x
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   
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L5
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9:.0_#5
\P
[ ]
t ∈ −
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
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t
f t m f t
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P
Giải. ⇔
( )
Z 
 Z
  [
x
f x x x x m m
≤ ≤



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
L5
( )
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]
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Z [ [ 8Z
x x x
f x
x x x
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∈
≥ −
⇔
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[ 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

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

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

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

( )
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n n
n n
x x x
u x x u x
n
n
x x x
v x x v x
n
n
−
−

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
−


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( )
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n n
x x x x
n
n
−
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9:.0_#5
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x−∞+∞f′+−
f
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⇒Q
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Z Z [ [

+ +
≥ ⇔ = ≥
+ +
+

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Z
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Z
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Z
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Z
Z
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t t t t
t
−
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[
Z

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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 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− + − =