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1. h .

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x ,

( ).

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e 1

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1

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x

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x

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x(t) M

y(t), x t 0 t 0 .

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g . 7

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x x0

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e 1 elimf(x) lim lim 1 f(0)

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,

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x x

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h f .

h xh (x) xe .

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h (x) 0 x 0 h (0, ) ,

h (x) 0 x 0 h (0, ) .

x 0 h(x) h(0) f (x) 0 .

x 0 h(x) h(0) f (x) 0 .

, f (x) 0 x ( ,0) (0, ) f 0x 0 , -

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0 0x x x0 0

2x 0 x 0 x 0 x 0

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f (0)2

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1

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x 0 .

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x

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r (x) 0 x 0 r (x) 0 x 0 r (0, ) .

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1.

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x

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x

x x

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4.

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1 1

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,

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e 1

1.

:

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xe x 0 .

2.

2 ( f )

f .

, f ( , )

x x

f , lim f x , lim f x 0,

x

x x

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x

0

x x0

x x x

e 1 elim f(x) lim lim

x 1

.

2.

2 ( )

0 .

, . f f () f () 2f () 2f () .

13

2f () 2f ()2f ()

1 1 2f ()

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,

( ) f(x) 0 x .

2.

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f ). F(x) .

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2f (x)

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F(2f (x)) F(1) 2f (x) 1 x 0

2.

4 ( )

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1

h(x) f(u)du, x ,

x

1

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f . f '' x . x 2

3 3

2x 2)e (x k(x)f '' )

2(x

x x

, x 2k(x) e (x 2x 2) 2 .

2 xk'(x) x e 0 x 0 . k

. 0 k , .

x 0 k(x) k(0) k(x) 0

x 0 k(x) k(0) k(x) 0 .

k(x) 3x f ''(x) 0 -

x .

h'(x) 2f 2f '(x) f ''(x) 0 x

h 0x 0 , h

x 0 ,

.

2.

5 ( )

x

1

T(x) f(u)du, x

14

1 1 .

Rolle T(x) [1, ] (1,)

T 0 f() 0 , ( f(x) > 0) 1 .

2f (x)

1

f(u)du 0

1

2f (x) 1 f x f x f 0 x 02

,

f "1-1" f . 2.

6 ( )

2x

1

F(x) f(u)du, x F (x) 2f(2x) 0 .

1 2x ,x 1 2x x .

f' ( ) 1 2f (x ) f (x )

F

1 22f (x ) 2f (x )

1 2

1 1

F(f (x )) F(f (x )) f(u)du f(u)du

2f (x)

1

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x 0 .

2.

2 ( )

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x(t)e 1

y(t) f x(t)x(t)

,

x(t) x(t) 2y (t) e x (t)x(t) e 1 x (t)x (t) 0t t

0 0 00

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x(t ) x(t ) x(t ) 20 0 0 0 0 0 0

x (t ) 0x(t ) x(t ) 2

0 0

x (t ) 2y (t ) x (t ) 2e x (t )x(t ) e e 1 x (t )x (t )

2e x(t ) 2e 2 x (t ) 0 (1)

x x 2k(x) 2xe 2e 2 x

xk (x) 2x e 1 .

k (x) 0 x 0 .

k (x) 0 x 0 .

k(x) -

0 . k(x)

0x 0 .

0x(t ) 0 .

15

0x(t ) 0 . 0 0y(t ) f(x(t )) 1 0

0

x'(t )1y (t )

2 2 .

0x(t ) 0 0 0y(t ) f x(t ) f(0) 1 A(0,1) .

2.

( )

, -

x(t) , -

t -

A(0,1) . tx(t) e 0x x(0) 1 0

tx'(t) e 0 t 0 , t x(t) 0 .

3.

2

g x g 1 g 2 0 x 0, , g ( )

1x 1 2x 2 .

g [1,2] 3x 1,2 -

, 3g x g x x 1,2 . 3x 1,2 , g x 0

x 1,2 , . , 3x 1,2 , g 3x .

3.

3

Rolle

g [1, 2].