0

: . . 0

2012

: ,

1

: . . 1

1. , :f g \ \ ( ) 5( ) 1 ( ) (1)f g x x x g x= + + + x\ . ) g 1-1.

) 0x \ , ( )0 0f x x= . ) f g ,

g xx.

) ( ) ( )1 2 1 2g x g x x x= = ( ) ( ) ( ) ( ) (1)1 2 1 25 5

1 1 1 2 2 25 5

1 1 2 2

( ) ( )

1 ( ) 1 ( )

1 1 (2)

g x g x f g x f g x

x x g x x x g xx x x x

= = + + + = + + +

+ + = + +

: 5( ) 1x x x = + + 4( ) 5 1 0 ( ) ( ) "1 1"x x x x = + > /

"1 1"

1 2 1 2(2) ( ) ( )x x x x = =

) ( )x

[ ] ( ).( 1) (1) 0( 1) 1

1,1 : ( ) 01,1(1) 3

2

: . . 2

2. , :f g \ \ : (1) (2) ... (2006) (1) (2) ... (2006) (1)f f f g g g+ + + = + + + . [ ]0 1, 2006x , : 0 0( ) ( )f x g x= .

( ) ( ) ( ) ( )1 (1) (1) (2) (2) ... (2006) (2006) 0 (2)f g f g f g + + + = : ( ) ( ) ( )h x f x g x= [ ]1,2006 , - [ ]( ) , 1, 2006m h x M x

[ ][ ]

[ ]

( ) (2)

1 1, 2006 (1)

2 1, 2006 (2)2006 (1) (2) ... (2006) 2006

...2006 1, 2006 (2006)

m h M

m h Mm h h h M

m h M

+

+ + +

2006 0 2006 0m M m M 0 ( )h x , ( )h x ,

[ ]0 0 0 01, 2006 : ( ) 0 ( ) ( )x h x f x g x = = 3. f [ ],a

( ) ( ) 0f a f > . ( )0 ,x a ( )0 0f x = . [ ],x a ( ) ( ) 0f x f a .

[ ],x a ( ) ( ) 0f x f a , [ ], : ( ) ( ) 0a f f a <

[ ] ( ). .

1 1

( ) ( ) 0, : ( ) 0

,f f a

a ff a

< =

, [ ] [ ). .

2 2

( ) ( ) 0( ) ( ) 0, ( ) 0

,( ) ( ) 0f ff a f

fff f a

= <

f ( ),a , .

3

: . . 3

4. [ ]: ,f a \ : 2( )f > . : 3 33 ( )f x dx

< , 2( )f x x= .

3 3

3 :

32

3x x dx

= : 2( ) ( )g x f x x= [ ]0 0, : ( ) 0x a g x =

2 2( ) ( ) 0 ( ) 0f f g > > >

3 33 3

2 2

3 ( ) ( )3

( ) ( ( ) ) 0

( ) 0

af x dx a f x dx

f x dx x dx f x x dx

g x dx

< < < < .

f f f f f ( [ ],a )

( ) 0( )

( )f x

f x f x

4

: . . 4

( ) 0 ( ) ( )(1) ( ) ( ) ( ) ( ) (2)( ) ( )

f x f a ff a f f f af a f

= =

: [ ]( )( ) ,( )

f xg x af x

= ,g :

( )( )

( )2

2 2

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) (3)( ) ( )

f x f x f xf x f x f x f xg x g xf x f x

= =

(2)

( )( )( )

( ) ( )( )( )( )

f ag af a

g a gfgf

= ==

[ ]( ) ( ). . 0 0

,

, , : ( ) 0( ) ( )

Rolleg

g x g xg a g

=

=

( ) ( ) ( )( ) ( )

20

2(3)0 0 0

0 20

( 02

0 0 0 0 0

0

( ( ) ( ), : 0

(

( ( ) ( ) ( ) ( ) 0f x

f x f x f x x a

f x

f x f x f x f x f xx

>

=

= >

6. f \ ( ) ( ) 0 (1)f x f x+ + = , x\ , * \ . ( ) 0f x = .

(1)

(1) ( ) ( ) 0 ( ) ( )

(1) ( ) ( ) 0 ( 2 ) ( )

( 2 ) ( ) (2)

x x

x x

f x f x f x f x

f x f x f x f x

f x f x

+ + + = = + + + + = + = +

+ =

( ) ( )2(1) 2 2 0 ( ) ( 2 ) 0( ) ( 2 ) 0 ( 2 ) ( ) (3)

(2), (3) ( 2 ) ( ) ( 2 ) (4)

x xf x f x f x f x

f x f x f x f xf x f x f x

+ + = + = + = =

= = +

( )f x 2T = . 0 (0) (2 )x f f = = ,

5

: . . 5

[ ]( ) ( ). .

0, 2

0, 2 0, 2 : ( ) 0(0) (2 )

Rollef

f ff f

=

=

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

f x x f x f x xf x f x f x

= = + + = = +

f 2T = . , f ( )0, 2 , , 2 ( ).

( ) 0f x = , .

7. f [ ]1, 4 . (1) 1, (2) 2, (3) 3f f f= > < (4) 4f = , ( )0 1, 4x

0( ) 0f x = .

(1) 1(4) 4

ff

= = : ( ) ( )g x f x x= , ( ) ( ) 1g x f x =

(1) (1) 1 0(4) (4) 4 0(2) (2) 2 0(3) (3) 3 0

g fg fg fg f

= == == >= , \ : ( ) ( )f f = .

( )1 i A + , 1 1

11

1 1 1

11

: ( ) (1 ) ( )

( ) (1)( )

x x

xx

x e if x i e if x i

ee f x

f x

+ = + + = += ==

\

( )1 i A + ,

7

: . . 7

22 2

2 2 22

: ( ) (1 ) ( ) (2)( )

xx xex e if x i e f x

f x

= + = + ==\

1

1 2

2

1

1 2

2

( ) 1( ) ( )(1), (2)

( ) 1

x

x x

x

f xf x f xe

f x e ee

= =

=

: ( )( ) xf xg xe

=

2

( ) ( ) ( ) ( )( ) ( ) (3)( )

x x

x xf x e f x e f x f xg x g x

e e = =

1

2

11

1 22

2

( )( )( ) ( )

( )( )

x

x

f xg xe g x g x

f xg xe

= =

=

[ ]( )

1 2(3).

1 2 1 2

1 2

,( , ) , : ( ) 0 ( ) ( )

( ) ( )

Rolleg x xg x x g f fg x g x

= =

=\

10. :f \ \ ( ) ( ) 1 (1)f x f x = x\ . (0) 1f = , :

) ( ) ( ) 1f x f x = x\ ) ( ) xf x e= x\ ) fC : 1y x = +

) (1) ( ) ( ) 1 (2)x x

f x f x =

( )( )

(1), (2) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) 0

( ) ( ) 0( ) ( )

f x f x f x f xf x f x f x f x

f x f x f x f x

f x f x xf x f x c

= =

+ = =

=\

0 (0) ( 0) 1x f f c c= = = , ( ) ( ) 1 (3)f x f x = , ( ) 0, ( ) 0f x f x

8

: . . 8

) 1(3) ( )( )

f xf x

=

1

01 1

1(2) ( ) 1 ( ) ( ) ( )( )

0 : (0) 1

xf x f x f x x f x C ef x

x f C e C

= = == = =

\

( ) xf x e=

) ( )0 0, ( )A x f x . : 00( ) xf x e= , 0

0( )xf x e = .

( )1 0 0 0 0 0 0 0: ( ) ( ) ( ) ( ) ( )y f x f x x x y f x x f x x f x = = + , 2 : 1y x = + . 1 2

0

0 0

0 00 0

0 0 0 0

( ) 1 1 0( ) ( ) 1 1 0 1 0 1 1

x

x x

f x e xf x x f x e x e e e = = = + = + = + = + =

1y x= + ( ) ( )00, 0,1e = 11. f [ ],a ( ), ( ) ( )f a f = . ( )1 2, ,..., ,a ,

1 2( ) ( ) ... ( ) 0 (1)f f f + + + =

i (1) [ ],a

.

: x + =

9

: . . 9

1 1

1 1 1

x

x

x

x x x

= = + =

= = =

f .

f ...

( )

1 1

2 2

3 3

( ) ( ), : ( )

( 2 ) ( ), 2 : ( )

( 3 ) ( 2 )2 , 3 : ( )

...

( ) (1 , : ( )

f a f aa a f

f a f aa a f

f a f aa a f

f f aa f

+ + = + + + + = + + + + =

+ + = ( )1 )

+

( )1 2

1 2

( ) ( ) ... ( )

( ) 2 ... ( ) 1

( ) ( ) 0( ) ( ) ... ( ) 0

f f f

f a f f f a f f a

f f af f f

+ + + = + + + + + + + =

+ + + = = =

10

: . . 10

12. f [ ],a , ( ) 0f = . ( ) 0f > , [ ],a f

.

f f f f f ( ) 0( ) ( ) ( )( ) 0 lim 0 lim 0

f

x x

f x f f xfx x

=

> > >

( ) 0f xx > .

( )0 : , x > ( ) 0f xx > .

( ),x ( ) 0f x < 0x < , . ( )( ) 0 ,f x x < f , [ ]( ) ,f x x 0 .

13. 1x

x xI dxe

= + . : i)

1

x

x

xe xI dxe

= + ii) I =

i) : ( )x y x a y = = + x y dx dy dy dx= = = x y yx y y

= = == = =

( ) ( )( ) ( ) 11 1 1x y yy yx x y yI dx dy dy

e ee

= = = =+ + +

1 1 1

y x

y y x

y

y y ye y xe xdy dy dxe e e

e

= = =+ + +

ii) 1 21

1

xx

xx

x

x xI dxx x xe xe I dx

exe xI dxe

+

= ++ = += +

11

: . . 11

( ) ( ) ( )( ) ( )( ) ( ) [ ]

(1 )2 21

2 2

2 ( ) ( )

2 ( 1) ( 1)

2 ( ) 2 2

x

x

x x eI dx I x xdxe

I x x dx I x x x dx

I xdx

I x

I I I

+ = =+ = =

= + = + + = + + = =

14. : /2

0

( )( ) ( ) 4

x

x xxI dx

x x

= =+ .

: 2

x y dx dy dy dx= = =

0 02 2

02 2 2

x y y

x y y

= = = = = =

2

2

x y y

x y x

= = = =

, ( )( ) ( ) ( )0

/2

y

y y

yI dy

y y

= +( )

( ) ( )/2

0

y

y y

yI dy

y y

= + ( )

( ) ( )( )

( ) ( )

( )( ) ( )

/2

0 ( ) /2

0/2

0

( )2

x

x x x x

x xx

x x

xI

x x x xI dx

x xx I dx

x x

+= + + = += +

/2

02 2 0

2 4I dx I I

= = =

12

: . . 12

15. f \ , 2( )f x x > x\ .

21( ) 12

f x x x= + , \ .

: 21( ) ( ) 12

g x f x x x= +

( ) 0g x = \ . 2 2( ) ( ) 0f x x x f x x x > > \ \

3 3

( ) 0 ( ) 03 3x xf x x f x x

> > \ \

3

( )3xf x \ .

( ) ( ) 1g x f x x = + 2 2 2( ) ( ) 1 1 ( ) 1f x x f x x x x g x x x > + > + > + 2 1 0x x x + > \ ( 0 < )

( ) 0g x x > \ , ( )g x / \ .

3

( ) ( ) 3 33 00 ( ) (0) ( ) (0)3 3

xh x f x xx h x h f x f=

> > > /

( )3( ) (0) 0,3xf x f x > + +

3

lim3xx

+ = + , lim ( )x f x+ = +

3 3( ) 00 ( ) (0) ( ) (0)

3 3

h x xx h x h f x f< < < / 3

( ) (0)3xf x f < +

3

lim3xx

= , lim ( )x f x =

( )g x / \ ( )( ) ( ) lim ( ), lim ( )x xg g f x f x + = =\ ( )g x \ ( )g = \

0 ( )g ( ) 0g x = \ ( )g x .

13

: . . 13

16. f [ ],a ( )Zx x i f x= + , ( )Wx x i f x= + , [ ],x a . Im( ) Im( )Za Z= , ( )1 2, ,a , 1 2 1 2W W W W + = + .

Im( ) Im( ) ( ) ( ) (1)Za Z f a f = = :

( ) ( )1 2 1 2

1 1 2 2 1 1 2 2

1 2 1 2

1 2 1 2

1 2 1 2

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2 ( ) 2 ( ) 0 ( ) ( ) 0

W W W Wif if if if

f f i f f if f f f

f f f f

+ = + + + + = +

+ = + = + = + =

( )

( )

1 1. .

2 2

( ) ( )2, , : ( ) 2, , ,

22 2

( ) ( ), , ,22 2 , , : ( ) 2

2

af f aaa a a a ff a

aa a f ff a a a f

+ ++ + = ++ + + =

1

1 2

1

( ) ( ) ( ) ( ) ( ) ( )2 2( ) ( ) 2 2

( ) ( ) 02 2 0

a af f a f f f f af f

f f

++ + + + = = =

= = =

7. 0a > 1xx e a= , .

0 0x > , 01

0xx e a=

( )

( )

0 0

0 0

1/ 1/0 0

0 00 0

(ln 1)0 0

ln ln ln ln ln

1 lnln 1 ln ln 1

ln ln 1

x x

x x

x e a x e a

ax a xx x

a x x a e

= = + = + = = =

, 0 0(ln 1)x xe .

: (ln 1)( ) x xf x e = ( )0,fD = +

14

: . . 14

[ ](ln 1)

(ln 1) (ln 1)

(ln 1)

0(ln 1)

( ) (ln 1) (ln 1 1)

( ) ln

( ) 0 ln 0 ln 0 1x x

x x x x

x x

ex x

f x e x x e x

f x e x

f x e x x x

= = + =

( )

( )

1 ( ) (1)( ) (1)

1 ( ) (1)

f x

f x

x f x ff x f

x f x f

< >

/

0

1(ln1 1) 1 1min ( ) (1) min ( )f x f f x e ee

= = = = . 1min ae

= .

18. f ( )2 2 2 2( ) 36 9 ,f x a x a x x x= + + \ 0a . f 0x x= , :

) : ( ) 22 00 0 2722 18 9a xa x x a + = + ) 0 0x = , .

)

0 .

0 0

0

( ) 0 (1)Fermat

f xx f x

x

=\

( ) ( ) ( )( ) ( ) ( )

( ) ( )( )

0

2 2 2 2

2 2 2 20 0 0 0 0 0

12 2 2

0 0 0

22 0

0 0 2

( ) 72 2 9 9 9

( ) 72 2 9 9 9

0 72 2 9 9

722 189

x x

f x a x a x x a x x a

f x a x a x x a x x a

a x a x x a

a xa x xa

=

= + + + + = + + +

= + + + = +

) ( ) 220 272 00 2 0 18 0 0 09ax a a = + = =+ , .

15

: . . 15

( ) 22 00 0 22 2 2

0 202 2 2

722 18 1 19

72 9 91 0 9 09 72 72

a xa x xa

a x a ax a aa a a

+ ++ + + + \

19. a, b, c, d :

(1) 0a b c d + + < (2) 0d > (3) 0a b c d+ + + < 2 3b ac . (: 3 2( )f x ax bx cx d= + + + ( )1,0 ( )0,1 ).

3 2

2

( )( ) 3 2

f x ax bx cx df x ax bx c

= + + + = + +

( 1) 0( 1) (0) 0

(0) 0

(0) 0(0) (1) 0

(1) 0

f a b c df f

f df d

f ff a b c d

= + + < = >

[ ] ( )[ ] ( )

. .

1 1

. .

2 2

1,01,0 : ( ) 0

( 1) (0) 0

0,10,1 : ( ) 0

(0) (1) 0

ff

f f

ff

f f

= < = . ) ( ) 0

x

xf t dt = .

) 0

( ) ( )x

f t f t dt ( ) ( ) 0 ( )F x f x x F x = > \ / \

) 0 0

( ) 0 ( ) ( ) 0x x x

xf t dt f t dt f t dt

= + =

( ) ( ) 0 ( ) ( )0

FF x F x F x F x

x x x x x x

+ = = = =

/

\

0x =

17

: . . 17

( ) 0

( ) 0f tx

xf t dt x x

>= =

22. f : ( ) 32( ) 2 3 33x xf x x x e x= + + ) f :

( ) 0f x > x\ . ) ( , )M a =

2

2

2

4( ) 0

y y

x xf t dt

=

) 2 2( ) (2 2) ( 2 3) 1x xf x x e x x e x = + + 2 2 2

2

( ) ( 1) ( 1) ( ) ( 1) ( 1)( ) 0 ( 1) ( 1) 0 1 0

x x

x x

f x e x x f x e xf x e x e x

= + + = + +

( ) ( ],0 ( ) 0 ( ) ,0x f x f x < 0 ( ) [ )0, ( ) 0 ( ) 0,x f x f x + > +/

0 ( ) (0) ( ) (0) ( ) 6( ) 0

0 ( ) (0)

f

f

x f x f f x f x f x x f x x

x f x f

> < >

/

2\ \

\

) 2

2

( ) 02 2 2

4( ) 0 2 4

f ty y

x xf t dt y y x x

> = =

( )

2 2

2 2

4 2 04, 2, 0

4 16 4 20 0

, 2,12 2

20 52

x y x yA BA B

A BK K

R R

+ == = =+ = + = >

= =

18

: . . 18

23. 1) :f \ \ : ( ) ( ) ( ) xf x f x f x ce = = , c .

2) ) \ f :

( )2 21 ( )( ) 1 1 (1)1x f tf x x dt xt = + + + \ ) f

(c) .

) (c)

.

1) ( ) ( ) ( ) ( ) 0 ( ) ( ) 0x x xf x f x f x f x f x e f x e e = = =

2 2

( ) ( ) ( ) 0 ( ) 0( ) ( )

( ) ( )

x x

x x x

xx

f x e f x e f x xe e e

f x c f x c e xe

= = = =

\

\

: ( ) ( ) ( )xf x ce f x f x = =

2) 2

2 21

21

( )( ) ( ) 1(1) 1

( )1 11

x

x

f t f x f t tdtf tx t dt

t

+ = + + + +

2 2 2 21

(1)

2

1 1 1

1 1 22

( ) ( ) ( ) ( )11 1 1 1( )

1(1) 1 (1) (1 1)(1 0) (1) 2

(1) 21 1 2

( ) ( ) ( 1)1

x

x

x x

f x f t f x f xdtx t x x

f x c exx f ff c e c e c e

f x e e f x e xx

= + = + + + + = +

= = + + == = =+= = ++

1 2 1 1 2

1 2

( ) ( 1) 2 ( ) ( 1 2 )( ) ( 1) 0

x x x

x

f x e x e x f x e x xf x e x x

= + + = +

= \

19

: . . 19

1x = , ( )f x / \ .

1 2 1 1 2

1 2

2 2

( ) ( 1) 2( 1) ( 2 1 2 2)( ) ( 1)

( ) 0 1 0 1 1 1

x x x

x

f x e x e x e x x xf x e x

f x x x x x

= + = + +

=

( )( )

1, ( 1) .

1, (1) .

f

f

: ( ) ( 1) ...B AA AB A

y yAB y y x x y fx x = =

.. , f , ( ) 0fC AB f x AB

( )1 11 1

( ) ( ) ...E f x AB dx f x AB dx = = =

24. f : 1

( )tx ef x dt

t= ,

0x . 1) x ( ) lnf x x .

2) ( ) : 1

limt xx

x

e dtt

+ .

1) : ( ) ( ) lng x f x x= . ( ) 0g x . 1 1 1( ) ( )

x xe eg x f xx x x x

= = = (1) 0g = ( ) 0 0g x x

1 ( ) (1) ( ) 01 ( ) (1) ( ) 0

x g x g g xx g x g g x < < <

( )g x 1x .

20

: . . 20

2) 11 1 1

( )lim lim lim lim lim

txt x t tx x xx x

x xx x x x x

e dte e e f xtdt e dt e dtt t t e e

+ + + + += = = =

1 1

1 ( ) ln

t t tx xe e e dt dt t t t

x f x x

lim ln lim ( )

x xx f x + += + = +

, ( ) ( ) 1lim lim lim lim 0( )

x

x x xx x x x

ef x f x xe e e x

+ + + += = = =

25. f [ ]0,1 , ( )1 1 10 0 0( ) ( ) ( )f x f y dy dx f x dx .

( ) ( )1 1 1 1 1 10 0 0 0 0 01 1 1

0 0 01 1 1 1

0 0 0 02

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ( ) )

f x f y dy dx f x dx f x f y dy dx f x dx

f y dy f x dx f x dx

f x dx f x dx f x dx f x dx

= == == = == =

1 0a = > , : 2( 1) 4 1 0 1

4 4 1 4a = =

26. f \ 3

( ) ( )x

xg x f t dt

+= , g \ .

3

0 0( ) ( ) ( )

( ) ( ) ( 3)

3 ( ) ( 3) ( 3) ( ) 0( ) 0 , ( )

x x

f

g x f t dt f t dt

g x f x f x

x x f x f x f x f xg x x g x

+= + = + +

< + > + + < <

2

\ 0 \

21

: . . 21

27. ( fC ) f

x 1 . ( fC ),

xx 1x = x a= : 2 1 2E a= + , 1a > .

1) f .

2) 1

lim ( )x

xxf t dt

++ .

1) ( )2 2 2 21 111 2 1 1 1a a a xE a E x x dx dxx = + = + = + = + :

2( )

1xf x

x= + \ .

2

2

1lim ( ) lim lim 11 1 01 1

x x x

x xf xx x

x+ + += = = =++ +

1y = +

2 2

1lim ( ) lim lim 11 11 1

x x x

xf xx

x x = = = + +

1y = .

2)

22

: . . 22

( )( )( ) ( )

( ) ( )

11 1 2 2

2 2 2 2

2 2 2 2

2 2

2 2

2 2

lim ( ) lim 1 lim 1

lim 1 1 1 lim 2 2 1

2 2 1 2 2 1lim

2 2 112

2 2 1lim lim2 2 11 1

xx x

x xx x x x

x x

x

x x

f t dt t dt t

x x x x x

x x x x x x

x x x

xx x x x

xx x x

++ ++ + +

+ +

+

+ +

= + = + = = + + + = + + + =

+ + + + + + += =+ + + +

++ + = = + + + +

2 22 2 11 1

2 0 2 121 0 0 1 0

xx x x

= + + + + += = =+ + + +

( )f t [ ], 1x x + ( )

( ) ( )[ ]

22 2 22 22

2 2 2 2 2

1 1 11( ) 01 1 1 1 1

( ) , 1

tt t t t t ttf tt t t t t

f t x x

+ + + + = = = >+ + + + + +/

- ( ) (1)m f t M , ( )m f x= , ( 1)M f x= +

( ) ( )1 1 1

1

1 1

(1) ( )

1 ( ) 1

( ) ( ) ( ) ( 1) (2)

x x x

x x xx

xx x

x x

mdt f t dt Mdt

m x x f t dt M x x

m f t dt M f x f t dt f x

+ + +

+

+ +

+ + +

( )

2 (2) 1

. .

2

lim ( ) lim ... 11

lim ( ) 11lim ( 1) lim 11 1

x xx

xx

x x

xf xx

f t dtxf xx

+ + ++

+ +

= = =+ =++ = =+ +

28. f \ ( )f x x < , x\ . : (4) (2) 6f f < .

23

: . . 23

2 2

( ) 0 ( ) 0 ( ) 02 2x xf x x f x f x

< <

24

: . . 24

(0) 3f = , 3c = : 21( ) 3 32

f x x x= + , 2x .

, f 2 , : 2

(2) lim ( )x

f f x= . , : ( ) ( )1, 2 2,3x , 2x

2

2 2

1(2) lim ( ) lim( 3 3) 12x x

f f x x x = = + = . : 21( ) 3 3

2f x x x + ,

x\ . , .

30B. f \ : ( ) 0f x

( )1 0 ( ) 2 2x u f t dt du x , x\ . fC , xx 0x = 1x = .

: ( )1 0( ) ( ) 2 2x ug x f t dt du x= + (1) 0g =

0( ) ( ) 2 (1)

xg x f t dt = ( ) 01 1 1

0 0 0( ) ( ) ( ) (2)

f xE f x dx f x dx E f t dt

= = = (1). 1

0

1 .1 ( ) (1) (1) 0 ( ) 2 0

1

Fermatg g g x g g f t dt

= =

\

(2)

2 0 2 . .E E = =

31. 4

1( ) 41

f xx

= ++ 0x > .

) f .

) 1

lim ( )x

xxf t dt

++ .

25

: . . 25

) 3

4

4( ) 0 02 1

xf x x fx

= > > + /

) ( )f t [ ], 1x x + f / - ( ( )m f x= ( 1)M f x= + )

( ) ( )1 1 1

1

1

( ) ( )

1 ( ) 1

( ) ( ) ( 1)

x x x

x x xx

xx

x

m f t M mdt f t dt Mdt

m x x f t dt M x x

f x f t dt f x

+ + +

+

+

+ + +

. . 1lim ( ) ...lim ( )

lim ( 1) ...xx

xxx

f xf t dt

f x

+++

+

= = + = ++ = = +

27

( )f x .

32. f 0 1x = , :

( ) ln 1 (1)f x x x , 0x > . : 1(1)2

f = .

1

f , : ( ) ( ) ln 1g x f x x x= +

(1) 0g = 1 1( ) ( ) ln ( ) (2)2

g x f x x f xx x

= +

( ) (1) 0g x g x >

( )(2).

11 1 11 (1) 0 (1) ln1 (1) 0 (1)1 2 2

1 0,

Fermatg

g f f f

= + = =

+

2

f , Fermat.

1 ln ln1 ln 0x x x> > >

26

: . . 26

00

1 1

1 1 1

( ) ln 1 1(1) ( ) (3)ln ln ln

11 12lim lim 1ln 2

1 1(3) lim ( ) lim lim ( ) (4)ln 2

x x

x x x

f x x x xf xx x x

x xx

xxf x f x

x+ + +

= =

1 ln ln1 ln 0x x x< < <

1 1 1

( ) ln 1 1(1) ( )ln ln ln

1 1lim ( ) lim lim ( ) (5)ln 2x x x

f x x x xf xx x x

xf x f xx

f 11 1

lim ( ) lim ( ) lim ( ) (1)xx x

f x f x f x f + = = = 1(4) (1) 12 (1)1 2(5) (1)2

ff

f

=

33. 2 2 21 2 2 3 1 3 (1)z z z z z z + = , 3 .

1 2 3( ) ( ) ( )A z B z z 2 2 2(1) + =

, , , , .

34. f , : , (1) 1 , ,f f z x = \ \ ^ \

( )2 11 1

2 5 ( ) 5 12 1 (1)x x tz i f t dt z i e dt x+ + +

) ( ) ( )M z c .

) ( )h x ( )c . )

1( ) ( )

xH x h t dt= , , , 1xx yy x = .

27

: . . 27

) ( )2 11 1

( ) 2 5 ( ) 5 12 1x x tg x z i f t dt z i e dt x= + + +

( )22

1 2

1

(1) 0

( ) 2 5 ( ) 5 12

( ) 2 5 ( ) 2 5 12 (2)

x

x

g

g x z i f x z i e x

g x z i f x z i x e

= = + + +

= + + +

(1) ( ) (1)g x g gC 1

. Fermat 2(2) 1 1(1) 0 2 5 (1) 2 5 1 12 0g z i f z i e = + + =

( ) ( )( ) ( ) ( )

5 5 6 5 5 6

0 5 0 5 6

0, 5 0,5

z i z i z i z i

z i z i

E E M z

+ + + = + + = + + =

6ME ME + = 2 6 3a a= =

( ) ( ) ( )

2 2 2 2 2 2

2 2 2 2

91 19 16 9 16 16 9

16 16 169 9 99 9 9

x y x y y x

y x y x y x

+ = = =

= = =

( )216( ) 99h x x=

) 1

0( )E H x dx=

[ ]

( )

1

1 1 11

00 0 01 1

1 00

1 0 1 1 1/22 2

1 1 0 0

*1/2

( ) ( ) ( ) ( ) 0 ( ) 0 ( )

1 ( ) (1) ( ) 0

( ) ( ) ( ) ( )

( ) ( )

4 41 ( ) 0 ( ) 9 93 3

2 23

x

x

H x h x dt H x h x H x H x

x H x H H x

E H x dx x H x dx xH x xH x dx

x h x dx xh x dx

h x dx h x dx x x dx x x dx

u

= = > > > > >

= = = = = =

= = =

=

/

8 8 81/2

9 9 9

2 2 ......3 3

xdx u du = =

28

: . . 28

2 9 2* 0 9

1 8

u x du xdxx ux u

= = = = = =

35. ( ): 1, , :f f + \ ( ) ( )0

2

4 2 ln 1 (1), 1x tf x t dt x x = + > . ) : ( )2

1( )1

f xx

= +

) g me ( )3 2( ) ( )g x x x f x= + , , , 1x x y y x = ) ( ) ( )x xh x g e e =

) 12 22

u x t du dt dt du= = =

( )

( )

0

02

0 0 0

0 0

(1)

0 0

0

0

2 02 20 2 0

4 12 ( )2 2

( ) ( ) ( )4 4 4

( ) ( )4 4

( ) ( ) ln 1

1( ) ( ) ( )1

( )

xx

x x x

x x

x x

x

x

x xt u x u

t u x u xxtf x t dt f u du

u x u xf u du f u du f u du

u xf u du f u du

uf u du x f u du x

xf x f u du xf xx

f u du

= = == = =

= = = = =

= + = +

2

1 1 1( ) ( )1 1 ( 1)

f x f xx x x

= = = + + +

) ( ) ( )23 2 2 21( ) ( ) ( 1) 0 1,( 1) 1xg x x x f x x x x x = + = + = > ++ +

29

: . . 29

( ) ( ) ( )

2 21 1 1

0 0 0

121

00

1 1( )1 1 1

1 ln 1 ln 1 ...2

x xE g x dx dx dxx x x

xx x dx x x

= = = + = + + + = + + = + + =

) + ( )2 0

0

0

lim ( ) lim ( ) lim1

lim lim1 1

1lim lim 111 11

, lim 0

lim lim 1

xx x x

xx x x

x x xx x

x x xx x

x x

xx x xx

x x

xx

xx u

eh x g e e ee

e e ee e le e ee e

e ee

u e u e

e ue u

+ + + +

+ +

+ +

+

+

= = =+= = =+ +

= = =+ + = = =

= =

1l = , lim ( ) 1x

h x+ = 1y = + .

36. :f \ \ 3 2( )f x x x x= + , x\ . i) f 1-1 ( )1,1A f 1f .

ii) fC ( )1,1A fC .

iii) 1f

( )1,1A . iv) :

1

1

( ) 1lim1x

f xx

.

i) 2( ) 3 2 1 0f x x x = + > x\ , 23 2 1x x + 4 12 8 0 = = < . , f 1-1. ,

30

: . . 30

(1) 1f = , 1(1) 1f = . ( )1,1A f 1f .

ii) ( ) fC ( )1,1A : (1) (1)( 1) 1 2( 1) 2 1y f f x y x y x = = =

( ) 2 1f x x= :

( ) ( ) ( )( )3 2 3 2

2 2

2 1 1 0

1 1 0 1 1 0

1 1

x x x x x x x

x x x x x

x x

+ = + = = = = =

( ) fC : ( )1, 3B iii) 1,f fC C y x= . ( ) fC ( )1,1A ( ) 1fC ( )1,1A .

( ) : 2 1y x= ( ) : 1 12 1

2 2x y y x= = +

iv) 1 1

1

( ) (1)lim1x

f x fx

1f 0 1x = ,

( ) , 12

= .

, 1 1

1

( ) (1) 1lim1 2x

f x fx

=

37. [ ): 0,f + \ . i) : ( ) ( 1) ( ) ( 1)f x f x f x f x < + < + , 0x > . ii) [ ), : 0,F G + \ :

1( ) ( ) ( )

x

xF x f t dt f x

+= 1( ) ( ) ( 1)xxG x f t dt f x+= + , 0x . iii) :

2 1

1 0(1) (0) ( ) ( ) (2) (1)f f f t dt f t dt f f < <

31

: . . 31

i) f [ )0,+ , ... [ ], 1x x + 0x > .

( ) ( 1) ( ), 1 : ( ) ( 1) ( )( 1)

f x f xx x f f x f xx x

+ + = = + +

( ) ( ) ( 1)f x f f x < < + , 0x > 1x x< < + . f / f .

ii) 1 1 1

1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )

x x x

xF x f t dt f t dt f x f t dt f t dt f x

+ += + = 1

1 1( ) ( ) ( ) ( 1)

x xG x f t dt f t dt f x

+= + , ( ) ( 1) ( ) ( )F x f x f x f x = + ( ) ( 1) ( ) ( 1)G x f x f x f x = + + 0x .

( ) 00

( ) 0F x

xG x

> > <

,F G [ )0,+ , F : G :

iii) ,F G : (0) (1)F F< (0) (1)G G> 1 2

0 12 1

1 0

(0) (1) ( ) (0) ( ) (1)

(1) (0) ( ) ( )

F F f t dt f f t dt f

f f f t dt f t dt

< < <

2 1

1 02 1

1 0

(1) (0) ( ) (2) ( ) (1)

( ) ( ) (2) (1)

G G f t dt f f t dt f

f t dt f t dt f f

< < <

: 2 1

1 0(1) (0) ( ) ( ) (2) (1)f f f t dt f t dt f f < < .

32

: . . 32

38. :f \ \ , : 20

1( )( ) 1

xf x dt

f t= + , x\ .

i) f .

ii) 0

( )limx

f xx

.

iii) : ( )3( ) 3 ( ) 3f x f x x+ = x\ . iv) f ( )f =\ \ , 1f 0, 1y x= = .

i) 21

( ) 1f t + \ 201

( ) 1x

f t + \ .

2 20

1 1( ) 0( ) 1 ( ) 1

xf x dt

f t f x

= = > + + x\ . , f .

ii) f 0

200

1lim ( ) (0) 0( ) 1x

f x f dtf t

= = =+ ( 00 :) de L 'Hospital

2 20 0 0

( ) ( ) 1 1lim lim lim 1( ) ( ) 1 (0) 1x x x

f x f xx x f x f

= = = = + +

iii) 21( )

( ) 1f x x

f x = + \

2 ( ) ( ) ( ) 1f x f x f x + = ( ) ( )2 33 ( ) ( ) 3 ( ) 3 ( ) 3 ( ) 3f x f x f x f x f x x + = + = 3 ( ) 3 ( ) 3f x f x x c+ = + x\

30 : (0) 3 (0) 0x f f c c= + = = , 3 ( ) 3 ( ) 3f x f x x+ = x\

iv) f / 1-1. 1f \ . 3 31( ) 3 3

3f x y y y x x y y= + = = +

, 1 3 21 1( ) 13 3

f x x x x x = + = + x\ .

33

: . . 33

1( ) 0f x [ ]0,1x . 13 4 21 11

0 00

1 1 7 7( ) . .3 12 2 12 2 12 12x x xE f x dx x dx E = = + = + = + = =

39. :f \ \ , : 20( )( ) 2

1x f uf x x duu

> + x\ . :

i) 202

( )1( )1

x f u duug xx+= +

, x\ .

ii) 20( ) 0

1x f uu

>+ 0x > . iii) ( ) 0f x > 0x > .

iv) 2 1

2 20 0

( ) 1 ( )1 2 1

x f u x f udu duu u

+>+ + , 1x > . v) lim ( )

xf x+ , + .

i) ( )

( )2

2 20 0

22

( ) ( )1 21 1( )

1

x xf u f ux du x duu ug x

x

+ + + = =+

( ) ( )( ) ( )

222 0 20

2 22 2

( ) ( ) ( )1 2 ( ) 211 1 01 1

xxf x f u f ux x du f x x duux u

x x

+ ++ += = >+ +

, x\ .

, g .

ii, iii) 0x > : ( ) (0)g x g>

202

( )1 01

x f u duux+ >+

, 2 1 0x + > 20

( ) 01

x f u duu

>+ 0x > : 2 0x > 20

( ) 01

x f u duu

>+ , 20

( )( ) 2 01

x f uf x x duu

> >+ , 0x >

34

: . . 34

iv) g/ 1x > : ( ) (1)g x g>

2 102 20

( )1 ( )1

1 2 1

x f u du f uu dux u+ >+ +

1x > :

2 1

2 20 0

( ) 1 ( )1 2 1

x f u x f udu duu u

+>+ +

v) (iv) : ( ) 122 20 0( ) ( )2 11 1x f u f ux dt x x duu u> ++ + , 1x > . ( ) 13 20 ( )( ) 1f uf x x x dtu> + + , 1x > (1)

, ( )3 3lim limx x

x x x+ ++ = = + 1

20

( ) 01

f u duu

>+ , ( ) 13 20 ( )lim 1x f ux x duu+ + = ++

( ) 13 20 ( )lim ( ) lim 1x x f uf x x x duu+ + + + , lim ( )

xf x+ = +

40. f , \ , [ ],a , 0a < < . : i) 1 2,x x , 1 2x x , 1 2( ) ( ) 0f x f x = = . ii) 3x \ , 3( ) 0f x = . iii) ( ) ( ) ( ) 0f x f x f x + = \ . iv) [ ]2( ) ( ) 0f x f x + = , \ .

i) f [ ],a , , 1x \ 1( ) (1)f x a= , 2x \ 2( ) (2)f x = . f \ , Fermat ( )1 0f x = ( )2 0f x = . , 1 2x x .

35

: . . 35

ii) .. 1 2x x< . f [ ]1 2,x x ( )1 2,x x 1 2( ) ( ) 0f x f x = = . Rolle, ( )3 1 2,x x x 3( ) 0f x = .

iii) ( ) ( ) ( ) ( )g x f x f x f x = + [ ]1 2,x x : 1 1 1 1( ) ( ) ( ) ( ) 0g x f x f x f x a = + = < 2 2 2 2( ) ( ) ( ) ( ) 0g x f x f x f x = + = >

Bolzano ( )1 2,x x ( ) 0g = . .

iv) ( )( ) ( ) f xh x f x e= [ ]1 2,x x ( )1 2,x x ( )2( ) ( ) ( )( ) ( ) ( ) ( ) ( )f x f x f xh x f x e f x e e f x f x = + = +

1( )1 1( ) ( ) 0

f xh x f x e= = 2( )2 2( ) ( ) 0f xh x f x e= = Rolle, ( )1 2,x x , : ( )2( ) 0 ( ) ( ) 0h f f = + =

41. ( )f x ( )0, + ( ) 1f x 2

21 1( ) 12

x

x

tf x f dtx x = + , 0x >

1( )( ) 1

g x xf x

= + , 0x > .

:

i) f .

ii) g .

iii) f ( )1

xf xx

= + , 0x > ( )f . iv)

0

( )limln( 1)x

f xx + .

i) : t u t xux= = dt xdu=

1t x u= = 2t x u x= =

36

: . . 36

[ ] [ ]2 21 1

1 1 1( ) 1 ( ) 1 ( )2 2

x xf x f u xdu f u du

x= + = +

, f : [ ] [ ]2 21

1( ) 1 ( ) 1 ( ) 0 (1)2

xf x f u du f x

= + = > , f .

ii) g : [ ](1)

21 ( )( ) 1 1 1 0

( ) 1 ( ) 1f xg x x

f x f x

= + = + = + =

iii) 1x = [ ]211

1 1(1) 1 ( )2 2

f f u du= + = 1 1(1) 1 1 2 1 11(1) 1 1

2

gf

= + = + = + = g , :

( ) 1g x = , 0x > . 1 1( ) 1 1 1 ( )

( ) 1 ( ) 1 1xg x x x f x

f x f x x= + = = = +

, 0

lim 01x

xx+

=+ lim 11xx

x+=+

, ( ) ( )00( ) lim ( ), lim ( ) 0,1xxf f x f x+ = = iv) ( )

( )20 0 0 0 0

11( ) 111lim lim lim lim lim 11ln( 1) ln( 1) 1ln( 1)1

x x x x x

xxxf x xx

x x xxx

++ += = = = =+ + ++ +

42. f , \ , : 22( )f x x z x z= + , z z z i= + , , , 0a a \ . ) lim ( ), lim ( )

x xf x f x+ .

) f , 1 1z z+ > . ) f .

37

: . . 37

( ) ( )2 22 222 2 22 2 2 2 2 2

4( )x a x a axf x x z x z x a i x a i

x a x a

+ + + = + = + = = + + + +

) 24 4lim ( ) lim lim 0

x x x

ax af xx x+ + +

= = =

2

4 4lim ( ) lim lim 0x x x

ax af xx x

= = =

) ( ) ( )2 22 21 1 1 1 1 1 0z z a i a i a a a + > + + > + + + > + >

f \ : ( ) ( ) ( )( )2 2 2 2 2 2

22 2 2( ) 4

x x a x x af x a

x a

+ + + + = + +

( )( )

( )2 2 22 2 2

2 22 2 2 2 2 24 ( ) 4 , 0

x ax aa f x a ax a x a

+ + + = = >+ + + +

( ) 2 22 2 2 22 , 0a aM f a Ma += + = >+ f :

( ) ( ) ( ) ( ) ( )( ] [ ] [ ) [ ]

2 2 2 2 2 2 2 2lim ( ), , , lim ( )

0, , ,0 ,x x

f f x f a f a f a f a f x

M M M M M M

+ = + + + + = = = \

f .

) 0a > : ( ) [ ] ( ) 2 2 2 22 2 2 22 2, ,a a a af M M f a a

+ += = + +

\ \

0a < : ( ) 2 2 2 22 2 2 22 2,a a a af a a

+ += + +

\

2 2 2

4( ) 0 0 0axf x xx a = = =+ + , f 0x = .

38

: . . 38

43. :f \ \ ( )20

11

f xdt x

t=+ ,

x\ . ) :

i) f 0 0x = (0) 0f = . ii) f \ .

) \ ( )( )( )( )2 2

23

1 01

fof

fofdt

t

+ >+

) f \ , (0) 1f = .

) i) (0)

20

10 01

fx dt

t= =+

2

1 01 t

>+ t\ , (0) 0f = .

(0) 0f > , (0)20

1 01

fdt

t>+ ,

(0) 0f < , (0)20

1 01

fdt

t+ + (1) 2( 2) (3 )f f + > f

2 22 3 3 2 0 1 2 + > + > < >

39

: . . 39

) ( )0 0 0

20

( ) (0) ( ) ( )lim lim lim 101

f xx x x

f x f f x f xx x dt

t

= =+

( )f x u= , 0

lim ( ) (0) 0x

f x f = = , 0u , 02

lim (2)11

uu

o

u

dtt

+

20

1( ) ,1

ug u dt u

t= + \ ,

0

2 20 00 0

1 1lim ( ) (0) lim 01 1

u

u ug u g dt dt

t t = = =+ +

(2) 00

De LHospital :

0 0 0

2 2020

( ) 1lim lim lim 11 11

1 11

uu u uu

u u

dtdtt u

t

= = = + ++

, (0) 1f = .

44. f \ (0) 1f = 2 1( ) 3 ( ) 1f x f x = + ,

x\ . . :

i) x\ : 3 ( ) ( ) 2f x f x x+ = + . ii) f 1f :

iii) f 0 2x = ( 2) 0f = . . f .

.

f , ,x x y y 2x = .

. i) 2 321( ) 3 ( ) ( ) ( ) 1 ( ) ( ) ( )

3 ( ) 1f x f x f x f x f x f x x

f x = + = + = +

3 ( ) ( )f x f x x c + = +

40

: . . 40

( )30 : ( ) ( ) 2 2x f x f x x c= + = + =

ii) ( )y f x= : 3 32 2,y y x x y y y+ = + = + \ . x, f 1-1 ,

: 1 3( ) 2,f x x x x = + \ .

iii) 1(0) 2f = : ( 2) 0f = .

. 21( )

3 ( ) 1f x

f x = + \ ,

: 226 ( ) ( )( )3 ( ) 1

f x f xf xf x

= +

( ) 0f x > f ( )f x . ( )f x -2. :

2 : ( ) ( 2) ( ) 0 ( ) 0x f x f f x f x< < < > 2 : ( ) ( 2) ( ) 0 ( ) 0x f x f f x f x> > > <

f ( )2,0 .

. [ ]0 02 2

( ) ( ) 2 ( ) 0E f x dx f x dx x f x = = : 1( ) ( )f x u x f u= = , : ( ) ( )1 3 2( ) 2 3 1dx f u du u u du u du = = + = + (0) 1f = , ( 2) 0f = . :

( ) 14 21 12 30 0

0

3 5(3 1) 3 . .4 2 4u uE u u du u u du = + = + = + =

41

: . . 41

45. f , 5( ) 5 ,f x x x a a= + \ .( ) ) f .

) f .

) ( ) 0f x = , 4 4a < < .

) 4( ) 5 5f x x = 4 4 4( ) 0 5 5 0 5 5 1 1f x x x x x = = = = =

f ( ], 1 [ )1,+ , f [ ]1,1 .

f 1 1x = : 5( 1) ( 1) 5( 1) 1 5 4f a a a = + = + + = + f 2 1x = : 5(1) 1 5 1 4f a a= + = .

) ( ]1 , 1 = f ( ) ( )5 5lim ( ) lim 5 lim

x x xf x x x a x = + = =

( 1) 4f a = + , ( ]1( ) , 4f a = + . ( )2 1,1 = f ( )

1lim ( ) ( 1) 4x

f x f a f + = = + ( )

1lim ( ) (1) 4x

f x f a f = = , ( )2( ) 4, 4f a a = + . [ )3 1, = + f (1) 4f a=

( ) ( )5 5lim ( ) lim 5 limx x x

f x x x a x+ + += + = = +

42

: . . 42

, [ )3( ) 4,f a = + . ( )( ) ,f A = +

) 4 0 4a a < < + , 10 ( )f , 20 ( )f 30 ( )f ( ) 0f x = 1 2, 3 . , ( ) 0f x = 3 \ .

46. f , \ 5 3( ) ( ) ( )f x f x f x x+ + = , x\ .

) f . ) f 1f . ) fC ( ),1A a ( ), 2B , fC x a= x = .

) : 4 25 ( ) ( ) 3 ( ) ( ) ( ) 1f x f x f x f x f x + + = 4 2

4 2

1( ) 5 ( ) 3 ( ) 1 1 ( ) 05 ( ) 3 ( ) 1

f x f x f x f xf x f x

+ + = = > + + ( )4 25 ( ) 3 ( ) 1 0,f x f x x+ + > \ . , f .

) f \ , f 1-1, f . x 1( )f x , : 5 3( ) ( ) ( )f x f x f x x+ + =

( ) ( ) ( )5 1 3 1 1 1 5 3 1( ) ( ) ( ) ( ) ( )f f x f f x f f x f x x x x f x + + = + + = , 1 5 3( )f x x x x = + + .

) ( ) ( )f

f a f a < , [ ],x a .

( ) ( )2 24 2 5 31 1

26 4 2 6 4 2 6 4 2

1

( ) 5 3 1 5 3

5 3 5 2 3 2 2 5 1 3 1 16 4 2 6 4 2 6 4 2

320 5 3 1 78312 2 . .6 6 4 2 12

E f x dx u u u du u u u du

u u u

= = + + = + + = = + + = + + + + =

= + + =

43

: . . 43

47. ) *,a z \ ^ z a z ai = + , Re( ) Im( )z z= . ) f [ ], ( ) ( )z f i f = + , ( )0 ,x , ( )0 0f x = .

) f ( ), 0x [ ], , fC .

) : z x yi= + , ,x y\

( ) ( )2 22 22 2 2 2 2 22 2 2 2 Re( ) Im( )

z a z ai x yi a x yi ai x a y x y a

x ax a y x y ay a ax ay x y z z

= + + = + + + = + + + + = + + + = = =

) ( ) ( )f f = ( ) f [ ], 2( ) ( ) ( ) 0f f f =

( ) 0f = , ( ) 0f = , ( ) ( )f f = . , f , ( ) ( ) 0f f < . Bolzano: ( )0 0, : ( ) 0x f x = .

) ... [ ]0, x [ ]0 ,x ( )1 0,x x , : 01

0

( ) ( ) 0 ( ) ( ) 2 ( )( ) (1)

2 2

f x f f f ff xx

= = = =+

( )2 0 ,x x , : 02

0

( ) ( ) ( ) 0 ( ) 2 ( )( ) (2)

2 2

f f x f f ff xx

= = = =+

o f [ ] ( )1 2, ,x x o 1 2( ) ( )f x f x =

44

: . . 44

Rolle: ( ) ( )1 2, , : ( ) 0x x f = , fC .

48. f \ , f y y ( )0,3A : 2( ) ( ) ( ) x xf x f x f x e e = + , x\ . ) ( ) 2xf x e= + . ) f 1f .

) 1fC , x x x e= 2x e= + .

) ( )2 22 ( )( ) ( ) ( ) ( )2 2x

x x xf x ef x f x f x e e f x e = + = +

2 2 2 2.

2 2

( ) ( )( ) ( )2 2 2 2

( ) 2 ( ) 2 2

x xx x

x x

f x e f x ef x e f x e c

f x f x e e c

= + = + + = + +

2 0 0 20 : (0) 2 (0) 2 2 3 2 3 1 2 2 3 3 2 0x f f e e c c c c= = + + = + + = + = , ( )2 2 2 2( ) 2 ( ) 2 ( ) 2 ( ) 2 0x x x xf x f x e e f x f x e e = + + =

( ) ( ) ( ) ( ) ( ) 222 2 2 22 4 1 2 4 4 8 4 2 1 4 1 2 1 0x x x x x x x xe e e e e e e e = = + + = + + = + = + > ( ) ( )2 2 1( ) 1 12

xx

ef x e

+= = +

, ( ) 2xf x e= + ( ) xf x e= , (0) 3f = ( ) 2xf x e= +

) ( ) 2xf x e= + ( ) 0xf x e = > , f \ , f 1-1, f .

( )2 2 ln 2 , 2x xy e e y x y y= + = = > , 1( ) ln( 2), 2f x x x = >

45

: . . 45

) 1( ) 0 ln( 2) 0 2 1 3f x x x x > > > >

( ) ( )( ) ( )

( ) ( ) [ ] [ ]

2 3 21 1 1

3

3 2

3

3 2

323 23

333 2

3

( ) ( ) ( )

ln( 2) ln( 2)

2 ln( 2) 2 ln( 2)

2 ln( 2) 1 2 ln( 2) 1

0 2 ln 2 0

e e

e ee

e

e

ee e

e e

e

e

E f x dx f x dx f x dx

x dx x dx

x x dx x x dx

x x dx x x dx

e e x e x

+ +

+

+

+ +

+

= = + == + =

= + == + + = = + +

( ) ( )2 ln 2 4 . .e e e

== +

49. f [ ],a , [ ]2,3 ( ) 2f a = , ( ) 1f = . ) ( )0 ,x a , : 0( ) 0f x = . ) f ( ),a , i) fC . ii) f ( ),a , ( ),a , : 2012( ) ( ) ( ) 0f f f + = . iii) ( )1 2, ,a , 1 2 , :

1 2

1 1( ) 2 ( ) 2

af f

= .

) -

min max2 ( ) 1 ( ) 2 3f f f a f = < = < = < = , : ( )1 1 min, : ( ) 2x a f x f = = ( )2 2, : ( ) max 3x a f x f = =

Bolzano [ ]1 2,x x [ ] ( )2 1, ,x x a , ( )0 0, : ( ) 0x a f x =

) i) 1 2,x x ( ),a f 1 2,x x

f 1 2,x x

Fermat: 1 2( ) ( ) 0f x f x = =

46

: . . 46

, fC ( )1 1, ( )A x f x ( )2 2, ( )B x f x .

ii) g , 2012( ) ( ) ( ) ( )g x f x f x f x = + g [ ]1 2,x x [ ] ( )2 1, ,x x a ( )201220121 1 1 1( ) ( ) ( ) ( ) 2 0 2 0g x f x f x f x = + = +

Bolzano: ( ),a , : 2012( ) 0 ( ) ( ) ( ) 0g f f f = + =

iii) ... [ ]0,a x [ ]0 ,x ( ) 0 01 0 1

0 0 1

( ) ( ) 0 2 1, : ( ) (1)( ) 2

f x f a x aa x fx a x a f

= = =

( )2 0 , :x 0 0

20 0 2

( ) ( ) 1 0 1( ) (2)2 ( ) 2

f f x xfx x f

= = =

( )0 0

1 2

1 1(1), (2)( ) 2 ( ) 2 2 2

x a x af f

+ + = + =

50. ,a \ , 2 24 9 1a + = z 2 1 2 31

z a iz

+ = + , 1z ,

* ` , : ) 1 1z + = ) z ) w 3 5 2w i+ = , w . ) z w+ .

) 2 1 2 31

z a iz

+ = + ,

2 1 2 31

z a iz

+ = +

47

: . . 47

( ) ( )

( )( ) ( )( )

( ) ( )( )( )

2 2 2 2

2 2

2

2 1 2 3 4 9 1 11

2 12 1 1 1 2 1 11 1

2 1 1 2 1 2 1 1 1

4 2 2 1 1

3 3 3 0 0

1 1 1 1 1

1 1 1 1 1 1 1

z a az

zz z zz z

z z z z z z

zz z z zz z z

zz z z zz z z

zz z z z z z

z z z z

+ = + = + = =

++ = = + = + = + + = + + + = + + + = + + = + + + = + + + = + + = + = + =

) z 1C ( )1,0K 1p = , z 2C ( )1,0 1p = .

) ( )5 3 2w i = , z 3C ( )5, 3M 2r = .

) ( )z w w z+ = w z . ( ) ( ) ( )

( ) ( )( ) ( )

2 25 1 3 5

max 5 1 2 8

min 5 1 2 2

z w r

z w AB r

= + =+ = = + + = + + =+ = = = =

48

: . . 48

51. f ( ),a lim ( ) lim ( )x a x

f x f x

+ = = .

) ( )( ), ,

( ),

f x xg x

x a x

= = =

.

) ( ),a : ( ) 0f =

) lim ( ) lim ( )

lim ( ) ( ),( )

x a x ax a

g x f xg x g a g

g a

+ +

+

= = == .

lim ( ) lim ( )lim ( ) ( ),

( )x x

x

g x f xg x g g

g

= = ==

.

f ( ),a , g ( ),a g ( ),a g a

gg

[ ],a .

) ( )( ) ( ) ( ), ( ) ( ) ,( )

g ag a g g x f x x a

g

= = = =

[ ]( ) ( )

,

, , : ( ) 0 ( ) 0( ) ( )

Rolleg

g a g fg a g

= =

=

52. 23

1

xa

xa

e xA dxe

+= + , (0) 0f = .

: 2 2 20

0

3 3 3( ) ( )1 1 1

x x xx x

x x xx x

e x e x e xf x dx f x dx dxe e e

+ + += = + + + + 2 2

0 0

3 3( ) (1)1 1

x xx x

x x

e x e xf x dx dxe e

+ + = ++ +

49

: . . 49

2 2

0

2 2

0 0

3 3 .1 1

3 31 1

x xx

x x

x xx x

x x

e x e xH dx e e

x

e x e xdx x dxe e

+ + + +

+ + + +

( )

( ) ( )( )

2 2

22

2 2

2 2 2 2

2 2 2

2 3

3( ) 3(1) ( )1 1

1 1 33 3 31 11 11

1 3 3 1 3 3( )1 1 1

(1 3 ) 1 3 1 3 11 1

( ) 1 3 ( )(

x x

x x

x

x xx x

xx x

x x

x x x x

x x x

x x

x x

e x e xf x xe e

e xx e x e xe eee e

e ee x e x e x e xf x

e e ee x x x e

e ef x x f x x x cf

+ + = + =+ +++ + += + = +++ ++

+ + + + + = + = =+ + ++ + + + += =+ +

= + = + +30) 0 0 0 0 0c c= = + + =

, 3( )f x x x= + 3

23

( )3

1

xa

xa

f a a ae x dx a a

e

= ++ = ++

53. ,f g ,f g

( )( ) ( ) ( ) ( ) ( ) ( )f x f x g x g x f x g x + = . : ( ) ( )f x g x= .

,f g , : 0 0 0: ( ) ( ) (1) x f x g x =\

50

: . . 50

( )( )

( ) ( ) ( )( ) ( )

0

2 2

2 2

2 2

2 20 0 0 0

(1)2 2

0 0 0 0

( ) ( ) ( ) ( ) ( ) ( )

2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( ) 2 ( ) ( )

x x

f x f x g x g x f x g x

f x f x g x g x f x g x

f x g x f x g x

f x g x f x g x

f x g x f x g x c

f x g x f x g x c

g x g x g x g x c c

=

+ = + =

+ = + =

+ = + + = + + = + 0=

, 2 2 2 2( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 0f x g x f x g x f x g x f x g x+ = + = ( )2( ) ( ) 0 ( ) ( )f x g x f x g x = =

54. f [ ]0,1 ( ) ( )22 1 ( ) ( ) (1)x f x x x f x [ ]0,1x . i) (0) (1) 0f f ii) f ( )0,1 .

i) (1) 0 : (0) 0 (0) 0 (0) (1) 01: (1) 0 (1) 0

x f ff f

x f f= =

ii) ( )f x ( )0,1 , ( ) [ ]( )( ) 0 0,1 ( ) 0, 0,1if x x f x x

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

f x f x

2

( )( )

x xg xf x= [ ]0,1 2( ) 0 (2)

( )x xg xf x

=

(0) 0(0) (1)

(1) 0g

g gg

= ==

51

: . . 51

[ ]( ) ( ) (2)0 00,1

0,1 0,1 : ( ) 0(0) (1)

Rolleg

g x g x g g

=

=

( )f x .

55. f [ ],a . : ( ) 1 21 2 3 ( ) 2 ( ), , , : ( ) 5f fa f + = .

[ ],a 5 5

... ( [ ]. . ,f f f ).

. . .

1

1 1

1

, 25

, 2 :5

, 25

2 ( ) 2 ( )5 5( ) ( )

2 25 5

2 ( )52 ( ) 5 (1)

f a aa a

f a a

f a f a f a f af f

a a

f a f af

+ + + + + = = +

+ =

52

: . . 52

( )

. . .

2

2 2

2 2

2 ,5

2 , :5

2 ,5

( ) 2 ( ) 25 5( ) ( ) 5 5 2 22

5 5

( ) 2 ( ) 25 5( ) 5 3 ( ) 5

3

f aa

f a

f f a f f af f

a

f f a f f af f

+ + +

+ + = = + + + = = (2)

1 2( ) ( )(1), (2) 2 ( ) 3 ( ) 5 (3)f f af f

+ =

[ ]( ) ( )

. . .

(3)

1 2

, ( ) ( ), : ( ),

2 ( ) 3 ( ) 5 ( )

f a f f aa ff a

f f f

= + =

!!!

: ( ) 1 2 3, : 6 ( ) 1 ( ) 2 ( ) 3 ( )a f f f f = + + 6 1 2 3= + + [ ],a

6

53

: . . 53

56. f \ , (1) (3)f f= ( ) ( )3 3f x f x x\ (1). ( ) 0f x =

.

: ( ) ( )3( ) 3h x f x f x=

( ) ( ) ( ) ( ) ( ) ( )3 3 2 3(1) (1) (3) (1) 0

( ) 3 3 ( ) 3 3 3 (2)

h f f h

h x f x x f x x h x x f x f x

= = = =

, ( ) 0 ( ) (1)h x h x h x \

. Fermat (2)

(1) 0 (1) (3)h f f = = [ ]( ) ( )1,3

1,3 1,3 : ( ) 0(1) (3)

Rollef

f ff f

= =

57. ( ) 1 2x xf x z i z x= + \ . 1, 1 1, , 1z i z z i z < < ) .

) ( ) 0f x = . ) ( )M z .

) 1z i z < , Re( ) Im( )z a i z z= + < . ) ( )lim 1x xx z i z

) : z i a = 0 1a< < 1z = 0 1< < , ( ) 2x x ff x a D= + = \ 1 2,x x \ 1 2x x<

54

: . . 54

1 2

1 1 2 2 1 1 2 2

1 2

1

1 2 ( )

1 21

1 2

2 2 ( ) ( )x

x

ax x

a x x x x x x x x

x x

x x a aa a a a f x f x

x x

+ + > + >

< >0

0

f 2 \ .

) : 0 0(0) 2 0f a = + = , 0 ( )f x f 2 f 1-1, 0

( )f x .

) ( ) ( ) 11 0 1 0,1 , 11 1z i

z i K Rz

< + < = <

(1 0 ) 1z i + < ( ) 21,0 , 1R = . , , ,

.

) ( )( ) ( )( )2 21 1 1 1z i z z i z z i z i z z < < + < ( ) ( )1 1

2 2 2 2 Im( ) Re( )

zz zi zi zz z z z z i z z

yii x y x y x z z

+ + < + < + < < > >

) : , 1z i a z = = (): 1 ,x xa a < < 2 2

( ) ( ) ( )1lim 1 lim lim 1 1 0xxx x x x xx x xz i z a a

>+

= = = + = + /

55

: . . 55

57)i) ln 1t t 0t > . ii)

2 1( )ln

x

xf x dt

t t= . f .

iii) f . iv) 0 ( ) , 0f x x x< < > . v) ln 2 ( )f x< 1x . vi) . i). ( )( ) ln 1, 0,g t t t t= + . g ( )0,+ ( ) 1 1'( ) ln 1 ' 1 tg t t t

t t= = = .

: '( ) 0 1g t t , g :

- ( )0,1 , - [ )1,+ , 1x = , (1) 0g = . ( ) (1) ln 1 0 ln 1g t g t t t t 0t > .

ii)H 1lnt t 0t > ln 0t t ( (i)).

: 0x > 2 0x > , ( )0,fD = + .

iii).. H 1lnt t ( )0,+ ,

2 1ln

x

xdt

t t ( )( )

22 1 ln(2 ) ln'( )2 ln(2 ) ln 2 ln(2 ) ln

x xf xx x x x x x

= = . ( )0,x + :

22

( ) ln( )

2 2'( ) 0 ln(2 ) ln 0 ln( ) 0 ln( ) 0

2 2ln( ) ln1 1 2f x x

xf x x xx x

xx x

=

/

, f : - ( ]0, 2 , - [ )2,+ , 2x = (2)f .

56

: . . 56

iv) ln 1t t ( ) 0t > , 2 1 0ln

x

xdt

t t> ,

( ) 0 (1)f x > . 1 :

ln 1 0ln

t tt t , ,

2 2 2 2ln 1 1 10 1 0 1 ( )(2)

ln ln lnx x x x

x x x x

t t dt dt dt dt x f xt t t t t t > > > >

(1) ,(2) : ( ) 0, 0x f x x< < > . v) 1t :

ln 1 10 0ln lnt

t t t t t , ,

2 2 21 1 1 10 ln(2 ) ln ( ) ln 2 ( )

ln lnx x x

x x xdt dt dt x x f x f x

t t t t t t < < < , 0 ( )f x x 0x > .

0 0lim 0 lim 0x x

x+ + = = ,

0lim ( ) 0x

f x+ = , f .

-

-

- ..

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