tính đơn điệu hàm số và cực trị của hàm số

Bài 4. Cực trị hàm đa thức
Chuyên đề 2
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:  !"#$%&'(1. 2.)*
+,-./0%1ƒ′x=∈a
b
CÁC BÀI TẬP MẪU MINH HỌA
Bài 1. !m
( ) ( )
2
3 4 2 5 6
5
mx m x m
y
x
+ + − −
=
+
785+∞
Giải: 9*785+∞⇔
( )
2
2
2 :
 5
5

mx mx
y x
x
+ +
′
= ≤ ∀ ≥
+

⇔
( )
2 2
2 :  2 : 5mx mx m x x x
+ + ≤ ⇔ + ≤ − ∀ ≥
 ⇔
( )
2
:
5
2
u x m x
x x
−
= ≥ ∀ ≥
+
( )
5
;
x
u x m
≥

⇔ ≥
<.=
( )
( )
2 2
: 2 2
 5
 2 
x
u x x
x x
+
′
= > ∀ ≥
+

⇒ux785+∞⇒
( )
( )
5
:
; 5
6
x
m u x u
≥
−
≤ = =
Bài 2. !m
( ) ( )

6 2
5
5 6 >
6
y x m x m x
−
= + − + + −
76
Giải. 9*?76 ⇔
( ) ( )
( )
2
2 5 6  6y x m x m x
′
= − + − + + ≥ ∀ ∈
5
@
( )
y x
′
A7(x=B*x=675⇔y′≥∀x∈86C
⇔
( )
[ ]
2
2 5 2 6 6m x x x x+ ≥ + − ∀ ∈
⇔
( )
[ ]
2

2 6
6
2 5
x x
g x m x
x
+ −
= ≤ ∀ ∈
+

5
Chương I. Hàm số – Trần Phương
[ ]
( )
6
;<D
x
g x m
∈
⇔ ≤
<.=
( )
( )
[ ]
2
2
2 2 E
 6
2 5
x x

g x x
x
+ +
′
= > ∀ ∈
+
⇒gx786C⇒
[ ]
( ) ( )
6
52
;<D 6
:
x
m g x g
∈
≥ = =
Bài 3. !m
( ) ( )
6 2
5
5 6 2
6 6
m
y x m x m x= − − + − +
7
[
)
2+∞
Giải: 9*?

[
)
2+∞
⇔
( ) ( )
2
2 5 6 2  2y mx m x m x
′
= − − + − ≥ ∀ ≥
5
⇔
( )
2
5 2 2 3 2m x x x
 
− + ≥ − + ∀ ≥
 
⇔
( )
( )
2
2 3
2
5 2
x
g x m x
x
− +
= ≤ ∀ ≥
− +


<.=
( )
( )
2
2 2
2 3 6

 2 6
x x
g x
x x
− +
′
= =
− +
5
2
6 3
6 3
x x
x x

= = −
⇔

= = +


F

( )
A 
x
g x
→∞
=

GHH⇒
( )
( )
2
2
;<D 2
6
x
g x g m
≥
= = ≤

Bài 4.
( )
( ) ( )
6 2 2
2 : : 2 5 2 6y x mx m m x m m= − − − + + − −

[
)
2+∞
Giải: 9*?7
[

)
2+∞
( )
2 2
6 2 2 : :  2y x mx m m x
′
⇔ = − − − + ≥ ∀ ≥
<  . 
( )
2
: 6 6m m
′
= − +V
(
)
2
6 6
: 
2 >
m
 
= − + >
 
 
 7 
y
′
=
 .  2  1
5 2

x x<
HIgx≥. 01JA*=
<.
( )
y x
′
≥
K
2x∀ ≥
⇔
[
)
2 G+∞ ⊂

( )
( )
2
5 2

4
5
4
2
2 6 2 6 2 6 4  5
2
3
2
2 6
m
x x y m m m

S m
m
′
∆ >


− ≤ ≤


′
⇔ < ≤ ⇔ = − + + ≥ ⇔ ⇔ − ≤ ≤




<
= <


Bài 5. !m
( )
2
2 5 5x m x m
y
x m
+ − + +
=
−
7
( )

5 +∞
Giải: 9*        7 
( )
5 +∞
 ⇔
( )
2 2
2
2 > 2 5
 5
x mx m m
y x
x m
− + − −
′
= ≥ ∀ >
−
⇔
( )
( )
2 2
 5
2 > 2 5  5
5

g x x
g x x mx m m x
m
x m



≥ ∀ >
= − + − − ≥ ∀ >
 
⇔
 
≤
− ≠
 


2
5
x
2
x
x2LM N
Bài 4. Cực trị hàm đa thức
Cách 1:Phương pháp tam thức bậc 2
<.=
( )
2
2 5 m
′
∆ = + ≥
,<g
x
=.21
5 2
x x≤


HIgx≥. 01JA*=
<.gx≥K∀x∈5+∞⇔
( )
5 G+∞ ⊂

( )
( )
2
5 2
5
5 
5 2 5 2 3 5  6 2 2
6 2 2
6 2 2
2 5
2
m
m
x x g m m m
m
S
m
′
≤
≤ ∆ ≥






⇔ ≤ ≤ ⇔ = − + ≥ ⇔ ⇔ ≤ −
≤ −





≥ +
= − ≤



Cách 2:Phương pháp hàm số
<.=g′
x
=>
x
−m≥>
x
−5O∀
x
O5⇒gx785
+∞
@.
( )
( )
( )
2
5

5 3 5 
6 2 2
; 
5 6 2 2
6 2 2
5
5
5
x
g m m
m
g x
m
m
m
m
m
≥



= − + ≥
≤ −

≥
  

⇔ ⇔ ⇔ ≤ −
⇔
  

≥ +

≤
  

≤
≤


Bài 6. !m
( ) ( )
2
> 4  2 6 6 5y m x m x m m= − + − + − +
P
x∀ ∈¡
Giải:Q7R*)
( )
4 >  2 6 y m x m x
′
⇔ = − + − ≤ ∀ ∈¡
( ) ( )
[ ]
4 > 2 6  5F5g u m u m u⇔ = − + − ≤ ∀ ∈ −
@
( )
[ ]
 5F5y g u u= ∈ −
A*
S7,
( )

( )
5 3 E 
>
5
6
5 2 2 
g m
m
g m
 − = − ≤

⇔ ⇔ ≤ ≤

= − + ≤


Bài 7. ! m   *  
5 5
  2  6
> T
y mx x x x= + + +
 ?  BU  V
x ∈¡
Giải: Q7R*)
5 5
  2 6 
2 6
y m x x x x
′
⇔ = + + + ≥ ∀ ∈¡

⇔
( ) ( )
2 6
5 5
 2 5 > 6 
2 6
m x x x x x+ + − + − ≥ ∀ ∈¡
( )
[ ]
6 2
> 5
 55
6 2
m u u g u u⇔ ≥ − − + = ∀ ∈ −
BU
[ ]
 55u x= ∈ −
<.
( ) ( )
2
5
> 2 2 2 5  F 
2
g u u u u u u u
′
= − − = − + = ⇔ = − =
WX"HH,<,7R*)⇔
[ ]
( )
( )

55
4
;<D 5
3
x
g u g m
∈ −
= − = ≤

6
5
x
2
x
Chương I. Hàm số – Trần Phương
Bài 8. N*
( ) ( ) ( )
6 2
5
5 2 5 6 2
6
y m x m x m x m= + + − − + +

!m%PY<*.'*Z>
Giải. [\ 
( ) ( ) ( )
2
5 2 2 5 6 2 y m x m x m
′
= + + − − + =

  @ 
2
: 6 m m
′
∆ = + + >
7
y
′
=
.21
5 2
x x<
]PY<*.'*
Z>
[ ]
5 2 2 5
F F F >y x x x x x
′
⇔ ≤ ∀ ∈ − =

5 m⇔ + >
B*
2 5
>x x− =
<.
2 5
>x x− = ⇔
( ) ( )
( )
( )

( )
2
2 2
2 5 2 5 2 5
2
> 2 5 > 6 2
53 >
5
5
m m
x x x x x x
m
m
− +
= − = + − = +
+
+
( ) ( ) ( ) ( )
2 2
> 5 2 5 6 2 5m m m m⇔ + = − + + +
2
: 35
6 : 5 
3
m m m
±
⇔ − − = ⇔ =
%^"BU
5 m + >
,<

: 35
3
m
+
=
B. ỨNG DỤNG TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ
I. DẠNG 1: ỨNG DỤNG TRONG PT, BPT, HỆ PT, HỆ BPT
Bài 1. JP" !=
4 6
5 6 > x x x+ − − + =

Giải. _0%1=
5
6
x ≤
_`
( )
4 6
5 6 > f x x x x= + − − + =

<.=
( )
> 2
6
4 6 
2 5 6
f x x x
x
′
= + + >

−
⇒f
x
7
(
5

6

−∞



;`%)f −5=7" !f
x
=.1',a
x
=
−5
Bài 2. JP" !=
2 2
54 6 2 Ex x x+ = − + +

Giải. Ha" !⇔
( )
2 2
6 2 E 54f x x x x= − + + − +
=5
Mb
2

6
x ≤
!f
x
c⇒5B$1
Mb
2
6
x >
!
( )
2 2
5 5 2
6 
6
E 54
f x x x
x x
 
′
= + − > ∀ >
 ÷
+ +
 
⇒f
x
7
(
)
2


6
+∞
*f 5=75.K51
x
=
5
Bài 3. JPa" !=
6 4
>
5 4 : : 4 56 : Ex x x x+ + − + − + − <
d
>
Bài 4. Cực trị hàm đa thức
Giải. _0%1
4
:
x ≥
_`
( )
6 4
>
5 4 : : 4 56 :f x x x x x= + + − + − + −
<.=
( )
( ) ( )
2 6 >
4
6
>

4 : 56
5

2 5
4 56 :
6 4 : > : 4
f x
x
x
x x
′
= + + + >
+
× −
× − × −
⇒f
x
7
)
4

:

+∞


;*f6=E7d⇔f
x
cf6⇔
x

c6
eX,1Y<a" !fA*
4
6
:
x≤ <
Bài 4. JPI=
6 2
5 5 5
4 > 6 2 2 4 : 5:
2 6 3
x x x x
x x x
x x x+ + + = + + − + − +
d
Giải. d
( )
(
)
( ) ( )
( )
6 2
5 5 5
4 > 6 2 2 4 : 5:
2 6 3
x x x
x x x x
f x x x x g x
⇔ = + + + − − − = − + − + =
<.fxB*g′x=−3x

2
+5x−:c∀x⇒gx

b1Y<fx=gxA**<Y<
( ) ( )
B*y f x y g x= =

@fx?FgxPB*
( ) ( )
5 5 56f g= =
7d.1',a
x

=5
Bài 5. !m;<D
( )
  5  2   2m x x x x x x
+ + ≤ + + + ∀
d
Giải. _` 
( )
2
2
     5  2t x x t x x x= + ≥ ⇒ = + = +
 ⇒
2
5 2t≤ ≤
⇒
5 2t≤ ≤
%.d⇔

( )
2
5 5 5 2m t t t t
 
+ ≤ + + ∀ ∈
 

⇔
( )
2
5
5 2
5
t t
f t m t
t
+ +
 
= ≥ ∀ ∈
 
+
⇔
( )
5 2
;
t
f t m
 
∈
 

≥
@
( )
( )
2
2
2

5
t t
f t
t
+
′
= >
+

7  f t       
5 2
 
 
 ⇒
( )
( )
5 2
6
; 5
2
t
f t f

 
∈
 
= =
 ⇒
6
2
m ≤
 ⇒
6
;<D
2
m =
Bài 6. JP" !
2 2
 
2E 2E  2
x x
x− =
2 2 2 2
  2 2  2  2
2E 2E   2E  2E 
x x x x
x x x x− = − ⇔ + = +
d
4
Chương I. Hàm số – Trần Phương
[\
( )
2E

u
f u u= +
<.
( )
2E A 5 
u
f u u
′
= + >
g,<
( )
f u

d
( ) ( )
2 2 2 2
     2 f x f x x x x⇔ = ⇔ = ⇔ =

> 2
k
x k
π π
⇔ = + ∈¢
Bài 7. !
( )
 x y ∈ π
/<f1
   
6 4 2
x y x y

x y
− = −


+ = π

Giải. 
       x y x y x x y y− = − ⇔ − = −

[\  *    `   
( )
( )
   f u u u u= − ∈ π
  <  .
( )
2
5
5 

f u
u
′
= + >

g,<
( )
f u
7
( )
π

].
( )
( )
>
6 4 2
f x f y
x y
x y
 =
π
⇔ = =

+ = π

Bài 8. JP1" !
6 2
6 2
6 2
2 5
2 5
2 5
x y y y
y z z z
z x x x

+ = + +

+ = + +



+ = + +

d
Giải. [\
( )
6 2
f t t t t= + +
BU
t ∈¡
⇒
( ) ( )
2
2
2 5 f t t t
′
= + + >
⇒ft
?
]$ah#+)P&
x
≤y≤z
⇒
( )
( )
( )
f x f y f z≤ ≤
⇒
2 5 2 5 2 5z x y z x y+ ≤ + ≤ + ⇔ ≤ ≤
⇒
x

=y=z=±
5
Bài 9. JP1a" !
2
6
6 2 5 
6 5 
x x
x x

+ − <


− + >


Giải.
2
5
6 2 5  5
6
x x x+ − < ⇔ − < <
_`
( )
6
6 5f x x x= − +
<.=
( ) ( ) ( )
6 5 5 f x x x
′

= − + <
⇒
( )
f x
PB*
( )
(
)
(
)
5 5 5
 5
6 2: 6
f x f x> = > ∀ ∈ −
II. DẠNG 2: ỨNG DỤNG TRONG CHỨNG MINH BẤT ĐẲNG
THỨC
Bài 1.
NiZ=
6 6 4

6j 6j 4j
x x x
x x x− < < − +
∀xO
Giải 
6

6j
x
x x− <

∀xO⇔
( )
6
 
6j
x
f x x x= − + >
∀xO
3
Bài 4. Cực trị hàm đa thức
<.
( )
2
5 
2j
x
f x x
′
= − +
⇒
( )
f x x x
′′
= −
⇒
( )
5  f x x
′′′
= − ≥
∀xO

⇒
( )
f x
′′
8M∞⇒
( ) ( )
 f x f
′′ ′′
> =
∀xO
⇒
( )
f x
′
8M∞⇒
( ) ( )
f x f
′ ′
>
k∀xO
⇒
( )
f x
8M∞⇒fxOfk∀xO⇒"

6 4

6j 4j
x x
x x< − +

∀xO⇔gxk
4 6
 
4j 6j
x x
x x− + − >
∀xO
<.g′xk
> 2
5 
>j 2j
x x
x− + −
⇒g′′xk
6

6j
x
x x− +
kfxO∀xO
⇒g′x8M∞⇒g′xOg′k∀xO
⇒gx8M∞⇒gxOgk∀xO⇒"
Bài 2.NiZ=
2
 
2
x
x x
π
 

> ∀ ∈
 ÷
π
 
Giải.
2  2
  
x x
x f x
x
> ⇔ = >
π π
∀x∈

2
π
 
 ÷
 
[\i*
2 2
 
 
 
g x
x x x
f x
x x
−
′

= =
lm,%h1gxkxx−x
<.g′xkx−xx−xk−xxc∀x∈

2
π
 
 ÷
 

⇒gxP7

2
π
 
 ÷
 
⇒gxcgk
⇒
( )
2
 

g x
f x
x
′
= <
∀x∈


2
π
 
 ÷
 
⇒f xP7

2
π
 
 ÷
 

⇒
( )
(
)
2
2
f x f
π
> =
π
⇔
2
  
2
x
x x
π

 
> ∀ ∈
 ÷
π
 
Bài 3.NiZ=
2 A A
x y x y
x y
+ −
>
−
∀xOyO
Giải. @xOyOAxOAy⇔Ax−AyO7#aS
i
:
Chương I. Hàm số – Trần Phương
⇔
5
A A 2 A 2
5
x
x y yx
x y
x
x y y
y
−
−
− > × ⇔ > ×

+
+
⇔
5
A 2
5
t
t
t
−
> ×
+
BU
x
t
y
=
O5
⇔
5
  A 2 
5
t
f t t
t
−
= − × >
+
∀tO5<.
( )

( )
( )
( )
2
2 2
5 > 5

5 5
t
f t
t
t t t
−
′
= − = >
+ +
∀tO5
⇒ft85M∞⇒ftOf5k∀tO5⇒"
Bài 4.NiZ=
5
A A >
5 5
y x
y x y x
 
− >
 ÷
− − −
 


( )
 5x y
x y

∀ ∈


≠


5
Giải. [\<%P?<m,=
MbyOx!5⇔
( )
A A >
5 5
y
x
y x
y x
− > −
− −
⇔
A > A >
5 5
y
x
y x
y x
− > −

− −
Mbycx!5⇔
( )
A A >
5 5
y
x
y x
y x
− < −
− −
⇔
A > A >
5 5
y
x
y x
y x
− < −
− −
[\*`ftk
A >
5
t
t
t
−
−
BUt∈5
<.

( )
( )
2
5 2 5
> 
5  5 
t
f t
t t t t
−
′
= − = >
− −
∀t∈5⇒ft5
⇒fyOfxyOxB*fycfxycx ⇒"
Bài 5.NiZ=
b a
a b<
∀aOb≥n
Giải. a
b
cb
a
⇔Aa
b
cAb
a
⇔bAacaAb⇔
A Aa b
a b

<

[\*`fxk
A x
x
∀x≥n
<.
2 2
5 A 5 A
  
x e
f x
x x
− −
′
= ≤ =
⇒fx8nM∞
⇒facfb⇔
A Aa b
a b
<
⇔a
b
cb
a

Bài 6. (Đề TSĐH khối D, 2007)
E
Bài 4. Cực trị hàm đa thức
NiZ

( ) ( )
5 5
2 2  
2 2
b a
a b
a b
a b+ ≤ + ∀ ≥ >
Giải. H#aSi
( ) ( )
5 5 5 > 5 >
2 2
2 2 2 2
b a
b a
a b
a b
a b a b
   
+ +
+ ≤ + ⇔ ≤
 ÷  ÷
   
( ) ( ) ( ) ( )
( ) ( )
A 5 > A 5 >
5 > 5 > A 5 > A 5 >
a b
b a b a
a b a b

a b
+ +
⇔ + ≤ + ⇔ + ≤ + ⇔ ≤

[\*`<B
( )
( )
A 5 >
x
f x
x
+
=
BU
x >
<.
( )
( ) ( )
( )
2
> A > 5 > A 5 >

5 >
x x x x
x
f x
x
− + +
′
= <

+
( )
f x⇒
P7
( )
( ) ( )
 f a f b+∞ ⇒ ≤
Bài 7. (Bất đẳng thức Nesbitt)
NiZ=
6
2
a b c
b c c a a b
+ + ≥
+ + +
∀abO5
Giải. ]$ah#+)P&a≥b≥_`xka⇒x≥b≥
O
<.5⇔f xk
x b c
b c c x x b
+ +
+ + +
BUx≥b≥O
⇒
( ) ( ) ( ) ( )
2 2 2 2
5 5
  
b c b c

f x
b c b c
x c x b b c b c
′
= − − > − − =
+ +
+ + + +

⇒fx8bM∞⇒
2
   
b c
f x f b
b c
+
≥ =
+
2
_`xkb⇒x≥OD\*gxk
2x c
x c
+
+
BUx≥O
⇒
( )
2
  
c
g x

x c
′
= >
+
∀O⇒gx8M∞⇒
6
   
2
g x g c
≥ =
6
G26,<
6
2
a b c
b c c a a b
+ + ≥
+ + +
∀abO
T
Chương I. Hàm số – Trần Phương
BÀI 4. CỰC TRỊ HÀM ĐA THỨC
A. CỰC TRỊ HÀM ĐA THỨC BẬC 3
I. TÓM TẮT LÝ THUYẾT
1. Hàm số: y=fx
( )
6 2
ax bx cx d a= + + + ≠

2. Đạo hàm:

( )
2
6 2y f x ax bx c
′ ′
= = + +
3. Điều kiện tồn tại cực trị
y=fx.o⇔y=fx.oB*o
⇔
( )
f x
′
=
.21"m1⇔∆′=b
2
−6acO
4. Kỹ năng tính nhanh cực trị
JP&∆′=b
2
−6acO%.
( )
f x
′
=
.21"m1
5 2
x x
BU
2
52
6

6
b b ac
x
a
− ± −
=
B**ox
5
x
2

np<<.)oY<*A*=
( ) ( )
2 2
5 5 2 2
6 6
F
6 6
b b ac b b ac
y f x f y f x f
a a
   
− − − − + −
= = = =
 ÷  ÷
   
^"x
5
x
2

A*B$q!)ofx
5
fx
2
hn
p<r"i" BU)hnX)<m,=
Bước 1:o1"\"<fxf′x<.=
( )
( )
( )
( )
2
5 2
6 T 6 6 T
b b bc
f x x f x c x d
a a a
 
′
= + + − + −
 ÷
 
<,
( ) ( ) ( ) ( )
f x f x q x r x
′
= +
BUX
( )
5r x =

Bước 2:@
( )
( )
( ) ( )
( )
( ) ( )
( )
2
5 5 5 5
5
2
2
2 2 2 2
2

6 6 T
7

2
6 6 T
b bc
y f x r x c x d
f x
a a
f x
b bc
y f x r x c x d
a a

 

= = = − + −
 ÷

′

=
 
 
 
′
=
 
 

= = = − + −
 ÷

 


Hệ quả:
_S+<oo." !A*=y=rx
_BU*#+)=y=fx
( )
6 2
ax bx cx d a= + + + ≠
!
S+<oo." !=
(
)

2
2
6 6 T
b bc
y c x d
a a
 
= − + −
 ÷
 
5
Bài 4. Cực trị hàm đa thức
II. BÀI TẬP MẪU MINH HỌA
Bài 1. !m*=
( ) ( )
6 2 2 2
5
2 6 5 4
6
y x m m x m x m= + − + + + + −

ox=−2
Giải:
( )
( )
2 2 2
2 2 6 5y x x m m x m
′
= + − + + +
⇒

( )
( )
2
2 2 2y x x m m
′′
= + − +
_*ox=−2!
( )
( )
( ) ( )
( )
2
2
2  > 6  5 6 
6
5 
2 

y m m m m
m
m m
y
m m

′


− = − + − = − − =
  
⇔ ⇔ ⇔ =

  
′′
− >
− >
  
− >



Bài 2. !a)*
( )
6 2
5
6 2
x x
f x ax= − + +
F
( )
6
2
6
6
x
g x x ax a= + + +

.)oZDn%r<
Giải:
( ) ( )
2 2
2 6 Ff x x x a g x x x a

′ ′
= + + = − +
<R!a<g′x.2
1"m1 
5 2
x x<
 B* f ′x . 2 1"m1 
6 >
x x<
 <
( ) ( )
( ) ( )
5 2
5 6 2 >
6 5 > 2 5 2
5 2
5
5 6  F 5 > 
>


a
a a
x x x x
x x x x f x f x
f x f x

′
<
∆ = − > ∆ = − >


< < <



 ⇔ ⇔



′ ′
< < < <

  ′ ′
<

d
<.=
( ) ( ) ( ) ( )
5 2 5 5 2 2
 6 2 6 2 f x f x g x x a g x x a
′ ′ ′ ′
   
< ⇔ + + + + < ⇔
   

( ) ( )
5 2
6 2 6 2 x a x a
+ + <
( )

( )
2
5 2 5 2
54
T 3 > > 54  
>
x x a x x a a a a
⇔ + + + = + < ⇔ − < <
Bài 3. !m
( ) ( ) ( )
6 2
2 6 5 3 2 5f x x m x m x= + − + − −
.S+<
N_NBUSy=ax+b
Giải:
( ) ( ) ( )
[ ]
2
3 5 2 f x x m x m
′
= + − + − =
⇔
( ) ( ) ( )
2
5 2 g x x m x m= + − + − =
9*.N_N⇔
( )
g x
=
.21"m1⇔

( )
2
6  6
g
m m
∆ = − > ⇔ ≠
o1"\"<fxgx<.=
( ) ( ) ( ) ( )
( )
2
2
2 5 6 6 6f x x m g x m x m m= + − − − − − +
eUm≠6!" !
( )
g x =
.21"m1x
5
x
2
B**
y=fxox
5
x
2
<.=
( ) ( )
5 2
g x g x
= =
7,<


( )
( )
( )
( )
( )
( )
2 2
2 2
5 5 5 2 2 2
6 6 6 F 6 6 6y f x m x m m y f x m x m m
= = − − − − + = = − − − − +
⇒_S+<N_NA*∆=
( )
( )
2
2
6 6 6y m x m m= − − − − +
55
Chương I. Hàm số – Trần Phương
<.∆BUSy=ax+b
⇔
( ) ( )
2 2
6 6F 

6
6 6
m m a
a

m a
m a m a
≠ ≠ <
<
 

⇔ ⇔
  
= ± −
− − = − = −

 
eX,ac!
6m a= ± −
Fa≥!%$mPf
Bài 4. !m
( ) ( ) ( )
6 2
2 6 5 3 5 2f x x m x m m x= + − + −
.N_NZ7
S'=y=−>x
Giải:<.=
( ) ( ) ( )
[ ]
2
3 5 5 2 f x x m x m m
′
= + − + − =

⇔

( ) ( ) ( )
2
5 5 2 g x x m x m m
= + − + − =
9*.N_N
( )
g x
⇔ =
.21"m1
( )
2
5
6 5 
6
g
m m
⇔ ∆ = − > ⇔ ≠
o1"\"<fxgx<.=
( ) ( ) ( ) ( ) ( ) ( )
2
2 5 6 5 5 5 2f x x m g x m x m m m= + − − − + − −
eU
5
6
m ≠
!" !
( )
g x =
.21"m1x
5

x
2
B**
y=fxox
5
x
2
<.=
( ) ( )
5 2
g x g x
= =
7,<
( )
( ) ( ) ( ) ( ) ( ) ( )
2 2
5 5 5 2 2
6 5 5 2 F 6 5 5 2y f x m x m m m y m x m m m
= = − − + − − = − − + − −
⇒_S+<N_NA*∆=
( ) ( ) ( )
2
6 5 5 5 2y m x m m m= − − + − −

_ooZ7S'=y=−>x!∆≡'
⇔
( )
( ) ( )
( ) ( )
( ) ( )

2
6 5 2 6 5 2 
6 5 >
5
5 5 2 
5 5 2 
m m
m
m
m m m
m m m


− − − + =

− − = −
⇔ ⇔ =
 
− − =
− − =



Bài 5. !m
( )
6 2
: 6f x x mx x= + + +
.S+<N_N
B$.BUy=6x−:
Giải: 9*.N_N⇔

( )
2
6 2 : f x x mx
′
= + + =
.21"m1
⇔
2
25  25m m
′
∆ = − > ⇔ >
o1"\"<fxf′x<.=
( ) ( ) ( )
( )
2
:
5 2
6 25 6
T T T
m
f x x m f x m x
′
= + + − + −
eU
25m
>
!" !
( )
f x
′

=
.21"m1x
5
x
2
B**
y=fxox
5
x
2
<.=
( ) ( )
5 2
f x f x
′ ′
= =
,<

( )
( )
( )
( )
2 2
5 5 5 2 2 2
: :
2 2
25 6 F 25 6
T T T T
m m
y f x m x y f x m x= = − + − = = − + −

52
Bài 4. Cực trị hàm đa thức
⇒_S+<N_NA*∆=
( )
2
:
2
25 6
T T
m
y m x= − + −
<.∆⊥y=6x−:⇔
( )
2 2
6 5
>4
2
25 6 5 25
T 2 2
m m m− = − ⇔ = > ⇔ = ±
Bài 6. !m*
( )
6 2 2
6f x x x m x m= − + +
.oo
Di<+<∆=
4
5
2 2
y x= −

Giải:9*.N_N⇔
( )
2 2
6 3 f x x x m
′
= − + =
.21"m1
⇔
2
T 6  6m m
′
∆ = − > ⇔ <
o1"\"<fxf′x<.=
( ) ( ) ( )
( )
2
2
5 2
5 6
6 6 6
m
f x x f x m x m
′
= − + − + +
eU
6m <
!" !
( )
f x
′

=
.21"m1x
5
x
2
B**
y=fxox
5
x
2
<.=
( ) ( )
5 2
f x f x
′ ′
= =
7
( )
( )
( )
( )
2 2
2 2
5 5 5 2 2 2
2 2
6 F 6
6 6 6 6
m m
y f x m x m y f x m x m
= = − + + = = − + +

⇒_S+<N_NA*'=
( )
2
2
2
6
6 6
m
y m x m= − + +

N)o
( ) ( )
5 5 2 2
  A x y B x y
Di<+<
( )
4
5
=
2 2
y x∆ = −

⇔'⊥∆sY<tHd<.
5 2
5
2
I
x x
x
+

= =
,<
d⇔
( )
( )
( )
2
2
2
2 5
6 5

6 2

4
2 5
5 
6 5 5
6 6 2 2
m
m
m
m
m m
m m

− × = −
=



 
⇔ ⇔ =
 
+ =
 

− × + + = × −


Bài 7. N
( ) ( ) ( )
6 2
2
 6 E 5 2 5
6
f x x a a x a x= + − − + +
1.N;u=9*A$.N_N
2.JP&*ox
5
x
2
N;u=
2 2
5 2
5Ex x+ ≤
Giải: 1. [\" !=
( ) ( ) ( )
2
2 2  6 E 5  2 f x x a a x a
′

= + − − + =
<.=
( ) ( ) ( )
2 2
2
 6 53 5  2  6 62 a a a a a a a
′
∆ = − + + = − + ≥ ∀
b
2 2
  6       a a a a a a a
′
∆ = ⇔ − = = ⇔ = ⇒ + =
B$Av
56
Chương I. Hàm số – Trần Phương
eX,∆′O∀a⇒f′x=.21"m1x
5
x
2
B**.N_N
2.nen<.=
( )
5 2 5 2
6  F > 5  2x x a a x x a+ = − = − +

( )
( ) ( )
2
2

2 2 2
5 2 5 2 5 2
2 6  E 5  2 T E 3 x x x x x x a a a a a a
+ = + − = − + + = + −
( )
( ) ( )
2 2
2 2
T T   6  5E 6  5Ea a a a a a= + + − + = − + ≤
Bài 8. N*
( ) ( )
( )
6 2 2
2
5 > 6
6
f x x m x m m x= + + + + +
1.!m*oha5O5
2.JV)oA*x
5
x
2
!;<DY<
( )
5 2 5 2
2A x x x x= − +
Giải: <.=
( ) ( )
2 2
2 2 5 > 6f x x m x m m

′
= + + + + +
1.9*oha5O5
( )
f x
′
⇔ =
.21"m1
5 2
x x
Pf=
5 2 5 2
5 5x x x x< < ∨ ≤ <
( )
( )
( )
( )
( )
( )
2
2
2
2 5 
3 : 
6 2 6 2

4 5
3 4 
2 5 
3 : 

6 2 6 2
5
5 5
2
2
f
m m
m
m
m m
f
m m
m
S
m
m
′

<


+ + <
∈ − − − +



′
∆ >






∈ − −
+ + <


⇔ ⇔ ⇔



 

≥

+ + ≥



∉ − − − +








<
< − +

< −


 





( )
4 6 2m
⇔ ∈ − − +
2.@
( )
( )
5 2
2
5 2
5
5
> 6
2
x x m
x x m m

+ = − +


= + +



⇒
( )
5 2 5 2
2A x x x x= − +
( )
2
> 6
2 5
2
m m
m
+ +
= + +
2
5
E :
2
m m= + +

( ) ( ) ( ) ( )
5 5
: 5 : 5
2 2
m m m m
−
= + + = + +
'
4 5m− < < −


⇒
( )
( )
2
2
T
5 5
T E 53 T >
2 2 2
A m m m
 
 
= − + + = − + ≤
 
 
eU
>m = −
!
T
;<D
2
A =
Bài 9. !m*
( )
6 2
5
5
6
f x x mx x m= − − + +
.%P)<)

N_B*NA*/a
Giải: @
( )
2
2 5 f x x mx
′
= − − =
.
2
5 m
′
∆ = + >
7f′x=.21
"m1x
5
x
2
B**ox
5
x
2
BU)oA*
( )
5 2
A x y
F
( )
2 2
B x y
o1"\"<fxf′x<.=

( ) ( ) ( )
( )
(
)
2
5 2 2
5 5
6 6 6
f x x m f x m x m
′
= − − + + +
@
( ) ( )
5 2
f x f x
′ ′
= =
7
5>
Bài 4. Cực trị hàm đa thức
( )
( )
(
)
( )
( )
(
)
2 2
5 5 5 2 2 2

2 2 2 2
5 5 F 5 5
6 6 6 6
y f x m x m y f x m x m= = − + + + = = − + + +
<.=
( ) ( ) ( )
( )
( )
2
2 2 2 2
2 2
2 5 2 5 2 5 2 5
>
5
T
AB x x y y x x m x x= − + − = − + + −
( )
( )
2
2
2
2 5 5 2
>
> 5 5
T
x x x x m
 
 
= + − + +
 

 
 

( ) ( )
(
)
2
2 2
> >
> > 5 5 > 5
T T
m m
 
= + + + ≥ +
 
 

⇒
2 56
6
AB ≥
eX,
2 56
;
6
AB =
DP,<⇔m=
Bài 10.!m*
( ) ( ) ( )
6 2

5 5
5 6 2
6 6
f x mx m x m x= − − + − +
o
x
5
x
2
Pf
5 2
2 5x x+ =

Giải: 9*.N_N⇔
( ) ( ) ( )
2
2 5 6 2 f x mx m x m
′
= − − + − =
.21
"m1⇔
( ) ( )
2

5 6 2 
m
m m m
≠



′
∆ = − − − >

⇔
3 3
5  5
2 2
m
− < ≠ < +
d
eU0%1d!
( )
f x
′
=
.21"m1x
5
x
2
B**fx
ox
5
x
2
nAven<.=
( ) ( )
5 2 5 2
2 5 6 2
F
m m

x x x x
m m
− −
+ = =
<.=
( ) ( )
5 2 2 5
2 5 2 5
2 2 6 >
2 5 5 F
m m
m m m
x x x x
m m m m m
− −
− − −
+ = ⇔ = − = = − =
( )
( ) ( ) ( )
6 2
2 6 >
2 6 > 6 2
m
m m
m m m m
m m m
−
− −
⇒ × = ⇔ − − = −
2

2
6
m
m
=


⇔
=


NP2)*,0Pf0%1deX,
5 2
2 5x x+ =
2
2
6
m m
⇔ = ∨ =
Bài 11.!m*
( )
6 2
5
5
6
f x x mx mx= − + −
ox
5
x
2

P
f0%1
5 2
Ex x− ≥

Giải: 9g.N_N⇔
( )
2
2 f x x mx m
′
= − + =
.21"m1
⇔
( ) ( )
2
  5m m m D
′
∆ = − > ⇔ ∈ = −∞ +∞
U
d
eU0%1*,!
( )
f x
′
=
.21"m1x
5
x
2
B**fx

ox
5
x
2
nAven<.=
5 2 5 2
2 Fx x m x x m+ = =
,<=
54
Chương I. Hàm số – Trần Phương
( )
22
5 2 5 2 5 2 5 2
E 3> > 3>x x x x x x x x− ≥ ⇔ − ≥ ⇔ + − ≥

2
> > 3>m m⇔ − ≥
2
5 34 5 34
53   
2 2
m m m
   
− +
⇔ − − ≥ ⇔ ∈ −∞ +∞
 ÷  ÷
   
U
Pfd
eX,

5 2
Ex x− ≥
!
5 34 5 34
 
2 2
m
   
− +
∈ −∞ +∞
 ÷  ÷
   
U
B. CỰC TRỊ HÀM ĐA THỨC BẬC 4
I. TÓM TẮT LÝ THUYẾT
1. Hàm số: y=fx
( )
> 6 2
ax bx cx dx e a= + + + + ≠
2. Đạo hàm:
( )
6 2
> 6 2y f x ax bx cx d
′ ′
= = + + +
3. Cực trị:[\
( )
f x
′
=

.
 






⇒













⇒

®óng 1 nghiÖm
cã ®óng 1 cùc trÞ
1 nghiÖm ®¬n
cã ®óng 2 nghiÖm
1 nghiÖm kÐp
cã 3 nghiÖm ph©n biÖt cã 3 cùc trÞ gåm C§ vµ CT


4. Kỹ năng tính nhanh cực trị
JP&f′
x
17B*#'ax=x

%.f
x
ox

BU
oA*
( )
> 6 2
    
f x ax bx cx dx e= + + + +
^"x

A*B$q!
ofx

^hnX)=
Bước 1:o1"\"<fxf′x<.=
( ) ( ) ( ) ( )

> 6 2
f x q x f x r x
′
= +
↓ ↓ ↓

BËc BËc BËc
Bước 2:@f′x

=7fx

=rx


Hệ quả:N)oY<*X>=y=fxZ7y=rx
II. CÁC BÀI TẬP MẪU MINH HỌA
Bài 4. !oY<*
( )
> 2
3 E 5y f x x x x= = − − −

Giải: <.=
( ) ( ) ( )
2
6
> 52 E > 5 2f x x x x x
′
= − − = + −
F
( ) ( ) ( )
52 5 5f x x x
′′
= + −
@" !
( )
f x

′
=
.51 x=2B*51%\"x=−5
7*.K5ox=2;`%)
( )
2 63 f
′′
= >
,<
( )
N
2 24f f= = −
eX,*.o
N
24f = −
B*%$.o
53
Bài 4. Cực trị hàm đa thức
Bài 5. N
( ) ( )
> 6 2
> 6 5 5f x x mx m x= + + + +
!mƒxq.o
*%$.o
Giải:
( ) ( ) ( )
[ ]
6 2 2
> 52 3 5 2 2 3 6 5f x x mx m x x x mx m
′

= + + + = + + +
F
( )
( ) ( )
2


2 3 6 5 
x
f x
g x x mx m
=

′

= ⇔

= + + + =

[\)%P?<m,=
a)b
( )
2
5 : 5 :
6 6 2 2  
6 6
g
m m m I
 
− +

′
∆ = − − ≤ ⇔ ∈ =
 
 
!
( )
2 g x x≥ ∀ ∈ ¡
⇔gx≥
x∀ ∈ ¡

g,<f′x17B*#'ax=*f′′=3m+5O∀m∈I
⇒
( )
N
 5f f= =
iA**q.o*%$.o
b)b
( ) ( )

5
 6 5 
g
m
g m
′
∆ >


⇔ = −



= + =

!
( )
( )
( )
2 2
2 2 3 > 6f x x x x x x
′
= − = −
( )
f x
′
=
⇔x=1%\"x=6
b!P7,<=
9*y=fxq.o
*%$.o
c)b
( )

5
 
g
m
g
′
∆ >



⇔ = −


≠

!f′x.61"m1
5 2 6
x x x< <
b!P7,<=
9* y = f x.o7
%$Pf,7R*)
Kết luận:
{ }
5 : 5 :
 5
6 6
m
 
− +
∈ −
 
 
U
Bài 6. N*
( ) ( ) ( )
> 6 2
6 2 5y f x x m x m x= = + + + +
NiZ=∀m≠−5*A$.o
x ≤

C§
( ) ( ) ( ) ( ) ( )
[ ]
( )
6 2 2
> 6 6 > 5 > 6 6 > 5 f x x m x m x x x m x m x g x
′
= + + + + = + + + + =
< .= 
( ) ( )
2
2
T 6 3> 5 T 5 5: 
g
m m m m m∆ = + − + = − + > ∀
 7 gx = .2
1"m1x
5
x
2

5:
x−∞6+∞f′−−+f
+∞
N+∞
x−∞x
5
x
2
x

6
+∞f′−+−Mf
+∞
N
N_
N+∞
Chương I. Hàm số – Trần Phương
nAven<.=
5 2
 5  5x x m m= + ≠ ∀ ≠ −
⇒I
( )
f x
′
=
.61"m1
x
5
x
2
[\2%P?<=
a)bmc−5!
5 2
 5 x x m= + <

⇒
5 2
x x< <
⇒HP7
b!HH,<

x =
C§
b)bmO−5!
5 2
 x x >

B*
( )
5 2
6 6

>
m
x x
− +
+ = <
⇒
5 2
x x< <
⇒HP7
b!HH,<
2
x x= <
C§
Kết luận:
eX,∀m≠−5*A$.
x ≤
C§
Bài 4. (Đề thi TSĐH khối B 2002)
!m*

( )
> 2 2
T 5y mx m x= + − +
.6o
Giải. Q7R*)
( )
( )
2 2
2 2 T 2  y x mx m x g x
′
⇔ = + − = =
.61
"m1
2
6
T

2
 6
m
m
m
m
< −

−
⇔ < ⇔

< <


Bài 5. !m
( )
> 2 >
2 2f x x mx m m
= − + +
.N_NAX"*<)0
Giải.
( )
( )
6 2
> > >f x x mx x x m
′
= − = −
<.=
( )
2
 f x x x m
′
= ⇔ = ∨ =

_*.N_N⇔
( )
f x
′
=
.61"m1⇔O
⇒61A*=
5 2 6
F  Fx m x x m= − = =
⇒6N_NA*=

( )
( )
( )
> 2 > > 2
 2 F  2 F  2A m m m m B m m C m m m m− − + + − +

⇒
>
F 2AB BC m m AC m= = + =

_tHNAX"*<)0
!
AB BC AC= =
⇔
>
2m m m+ =
5E
x−∞x
5
x
2
+∞f′−+−M f
+∞
N
N_
N+∞
x−∞x
5
x
2

+∞f′−+−Mf
+∞
N
N_
N+∞
x−∞x
5
x
6
+∞f′−+−Mf
+∞
t
N
HN_
NN+∞
Bài 4. Cực trị hàm đa thức
> >
6
> 6 6m m m m m m⇔ + = ⇔ = ⇔ =
Bài 6. NiZ=9*
( )
> 6 2
5f x x mx mx mx= + + + +
%$
.N_B*N
m∀ ∈ ¡
Giải. [\
( )
( )
6 2 2 6

> 6 2  6 2 5 >f x x mx mx m m x x x
′
= + + + = ⇔ + + = −

⇔
6
2
>
6 2 5
x
m
x x
−
=
+ +
[\*
( )
6
2
>
6 2 5
x
g x
x x
−
=
+ +
.[_=
g
D = ¡

( )
( )
( )
( )
( )
2
2 2 2 2
2 2
2 2
> 6 > 6 > 2 5 5

6 2 5 6 2 5
x x x x x x
g x x
x x x x
 
− + + − + + +
 
′
= = ≤ ∀ ∈
+ + + +
¡
F
( )
2
>
A A
2 5
6
x x

x
g x
x
x
→∞ →∞
−
= = ∞
+ +
b1Y<" !
( )
f x
′
=

wA**<Y<
Sy=mBUy=gx
b!P7,<Sy=m-y=gxK5
⇒
( )
f x
′
=
.K51
eX,*y=fx%$.oB*o
Bài 7. NiZ=
( )
> 6
f x x px q x= + + ≥ ∀ ∈ ¡
⇔
>

243 2:q p≥
Giải. <.=
( )
( )
6 2 2
> 6 > 6 f x x px x x p
′
= + = + =
⇔
6
>
p
x
−
=
B*1%\"x=
@f′xx'aBU>x+6p7AX"P7<.=
fx≥∀x∈Ρ⇔
( )
6
; 
>
p
f x f
−
 
= ≥
 ÷
 
⇔

>
>
243 2:
 243 2:
243
q p
q p
−
≥ ⇔ ≥
Bài 8. (Đề thi dự bị ĐH khối A năm 2004)
!m*
> 2 2
2 5y x m x= − +
.6oA*6qY<<
)B$m
Giải. 9*    .  6  o   
( )
2 2
> y x x m
′
⇔ = − =
 .61  "m  1
m⇔ ≠
%..6oA*
( )
( ) ( )
> >
5 F 5  5A B m m C m m− − −

@

y
A**y7QNH
  5AB AC m⇔ = ⇔ = ±
uuur uuur
5T
x−∞x
2
+∞f′−−f
+∞
−∞
Chương I. Hàm số – Trần Phương
Bài 9. NiZ=
( )
> 2
3 > 3f x x x x= − + +
A$.6o
zA*VmY<<)l6qA*6o
Bài 10. NiZ=
( )
>
f x x px q x= + + ≥ ∀ ∈¡
⇔
6 >
243 2:q p≥
Bài 11. N
( ) ( )
> 6 2
E 6 2 5 5f x x mx m x= + + + −
!mƒxq.o
*%$.o

2