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【2019年整理】3[1]33函数的最大(小)值与导数课件2_图文

3.3.3

1-1



: y=f(x)
f(x)
f(x)

y
y=f(x)

y

y=f(x)

f '(x)<0

f '(x)>0

oa

bx

oa

bx

f (x) 0 , f (x).


y

y x0

o

x0

x

o

x0

x

f(x)x0

X0f(x)<f(x0),

f(x0) f(x) y= f(x0)

X0f(x)>f(x0), f(x0) f(x)y= f(x0)

.

,
y

y=f(x)

o a x1 x2 x3 x4 x5

x6 b x

f (x1), f (x3), f (x5 )
y=f(x) f (x2), f (x4), f (x6) y=f(x)

1 2f'(x) 3f'(x)=0 4f'(x)=0

5f'(x)f'(x)=0 f(x)



,





1:
y=f(x)I M
1xIf(x)M; 2x0If(x0) = M My=f(x)
2: y=f(x)I
M 1xIf(x)M; 2x0If(x0) = M
My=f(x)

,

x(a, b)
.
xa,b


y

y=f(x)

o a x1 x2 x3 x4 x5

x6 b x

y

f(x)max=f(a), f(x)min=f(x3)

y=f(x)

o a x1 x2 x3 x4 x5

x6 b x

[a,b]
y

y=f(x)

o a x1 x2 x3 x4 x5

x6 b x

[a,b]y=f(x)



y

[a,b]

y=f(x)

y=f(x)

a x1 o X2

X3

bx

__f_(_x_1)___f(_x_3_) ___f_(_x_2)____ ___f(_b_) _ __f_(_x_3)__

f(x3)f(b)


f(x)[a,b] (1) f(x)(a,b)() (2) y=f(x)f(a)f(b)()
, .
: 1., ;
2..

,:
(1),, , ,.
(2)[a,b].(a,b) ,, .
(3), ,, ()().



1:f (x) 6 12x x333.

f ' x 12 3x2 x3,3

f ' x 0,x 2x 2
10

f (2) 22f (2) 10f (3) 15, f (3) 3

f (x) 6 12x x333

2

2210.

3


2:y=x4-2x2+5[-2,2] .

: y 4x3 4x.
y 0 ,x=-1,0,1. x, y, y :

x -2 (-2,-1) -1 (-1,0) 0 (0,1) 1 (1,2) 2

y'

- 0 + 0 - 0+

y 13 4 5 4 13

,13,4.

y = x+ 3 x9x [4 , 4 ]
76 , -5 .
: (1) f (x)=3x+6x9=0, x1=3x2=1
f (3)=27, f (1)=5 (2) [4 , 4 ] f (4) =20 , f (4) =76
xy y

x -4 (-4,-3) -3 (-3,1) 1 (1,4) 4

y

+ 0 - 0+ 0

y 20

27

-5

76

4 , 4 f (4) =76 f (1)=5


2f (x) 2x3 6x2 a22 37 1a 2f (x)22
:(1f (x) 6x2 12x f (x) 0x 0x 2 f (2 40 a, f (0) a, f (2) 8 a 40 a 37a 3
(2)(1)f (x)2, 23.




1aby=fx aba,b

2f(x)
3 f(x)a,b






f x=4x3



4x2



x



01 2



f '(x) 12x2 8x 1 (2x 1)(6x 1)

1

f (x)



f( ) 6



:
D 1( )
(A)

(B) (C)

(D),.

2. y=f(x)a,b M
A m, M=m, f ( x) ( )
(A) 0 (B) 0 (C) 0 (D)

A 3. y= 1 x4 1 x3 1 x2 ,1,1( ) 432

(A)0

(B)2 (C)1

(D) 13

C 12
4y=x3-3x224 )

(A) -4 (B) 0 (C) 16 (D) 20

1. f(x)=x2-4x+6[15]
:f(x)=x2-4x+6 : f '(x)=2x-4 f '(x)=02x-4=0 x=2

x 1 12 2 25 5

y

-

0

+

y3

2

11

f(x) [15]3 112

2f(x)

1 2

x



sinx[02

]

f (x) 1 cosx
2

f

(x)



0



x1



2,
3

x2



4
3

x

0 (0,

)2
3

2
3

f (x)

+0

f (x) 0

3 32

(2 ,4 ) 4 (4 2, ) 2

33

33

-0+

2 3



32

f ( x ) , 0.

( 200921T )

f x 1 x3 x2 m2 1 xx R,

m 0.

3

1 m 1 y f x

1, f 1

2 f x

:11; (2) f x 1 m,1 m, 1 m,1 m.

f

x





2 3

m3



m2



1 3

;

f

x



2 3

m3



m2



1 3

042112 af ( x) ( x2 4)( x a)
f ( x) f '(x) 3x2 2ax 4

f (1) 0 f ( x)[-22]

f ( x) --2][2+

a

a1 2

fmax



f

(1)

9, 2

fmin



f

(4) 3



50 27

f '(x) 3x2 2ax 4 0[ 2, 2]

2 a 2



. . .