2016

-

- 1 -

-

- 2 -

-

- 3 -

-

- 4 -

.

-

- 5 -

-

- 6 -

, ,

. 1 !

1 ( )

.133: ()

.141: ( )

.142: ( )

.143: ( + )

.144:

.149: ( , )

.150: ( )

.151: ( 1 -1)

.152: +

.153:

.153-154: ( )

.154: f - 1 ( f (x ) )=x, xA f ( f - 1 (y ) )=y, y f (A)

.155: .

.159:

.160:

.161:

.162:

.163:

.165: 1

.166:

.167: : 0

0

0x x

x x

0

0

limP(x) = P(x )

limP(x )P(x)

Q(x) Q(x )

.169: ( )

.170: ( )

171: ( )

.173:

0 0x x u u

limf g(x) = limf(u)

.178: ( )

.179:

.183: (

)

.184:

.185:

.186: ()

.188: ( x0 )

.189: f .

.

-

- 7 -

.190:

.191: ( )

.192: Bolzano (

)

.192: ( )

.194: ( )

.194: +

.195: ( ) +

.196: ( )

.201-03:

2 ( )

.212: ( C f )

.213: ( f x0 )

.213: + ,

.214:

.217: ( )

.218:

.222: . f .

.223: ( c ) =0 (x )=1

.224: : (x ) = x - 1 , 1

2 x

.226:

.229: ( )

.230: ( )

.230:

.231: ( )

.231: (x - )=-x - - 1

.232: (x )=1

2 x

+

.234: ()

.234: (x )= x - 1 ( x ) = x ln

.235: 1 ln x

.235:

.241: ( )

.241-242: , -- .246: Rolle + .246: .. + .251: .251: .252:

-

- 8 -

.253: .254:

.258: ( )

.259: ( ) -

.260:

260-1: (Fermat)

.261: ( - )

262:

.264:

.264: ( )

.273: (-)

.274:

.275: ( )

.275:

.276:

.279: ( )

.280:

.281:

.282:

.283:

.287:

.295-9:

3 ( )

.303: ( )

.304:

.328:

.329-330:

.330:

.332:

.334:

.334-5:

.336:

.337:

.342-345:

.346:

.348:

.354-9:

-

- 9 -

.

-

- 10 -

-

- 11 -

, , , ,

(

,

)

2 .226-227 , 3 . 247-248, .252, 2 .254-256 , 3 . 265-267 1 2 .346-347

-

- 6/148, 2/156

+ 2/176, 4/176, 3/182, 4/182, 1/187, 3/187, 4/187, 3/102, 1/285, 2/286

2/199, 3/199, 6B/286

Bolzano + . +

4B/199, 5B/200,8B/200

6/200, 4/257

7/200

9/200

x0 3A/220, 2B/220, 4B/220, 6B/221, 7B/221,

8B/221, 1B/228, 5B/286, 7/240

2/228, 3/228, 4/228, 5/238, 7/239, 10/239, 11/239, 1/240,

2/240, 3/240, 4/240, 6/240, 8/24011/241, 12/241

12/239, 14/239

1/244, 2/144, 4/244, 5/244

.Rolle .. 3/249, 1/249, 3/250, 4/250, 5/250,

6/250, 7/250

+ +

1/256, 1/257, 11/293, 4/308, 1/308, 3/309, 4/309, 11/351

2/257, 6/257, 2/291

7/258, 8/258, 3/269

6/256, 7/ 256, 5/257, 2/267,

1/269, 2/269

, 8/268, 4/269, 6/270, 8/270

. Fermat 5/268, 5B/270, 7/292

- 3/278, 4/278, 5/279, 8/292

+ +

1/338, 7/339, 8/339, 9/339, 3/338, 4/338, 6/339, 11/340,

12/340, 1/352, 2/352, 4/352, 7/353

6/352

-

- 12 -

10/353

1A/349, 2A/349, 3/349, 4A/349, 5/349, 1/349, 2B/349, 3/350, 4B/350, 5/350, 8/351, 9/351, 10/351, 12/351, 8/353,9/353

-

- 13 -

. -

-

- 14 -

-

- 15 -

-

1

f (x )=2lnx g(x) = lnx2 .

2

.

3

x = x x = 0

4

0 0

00

( ) ( )( ) lim

h

f x h f xf x

h

5

f x0 , .

6

f

7

2

0

2 = 0

8

f R :

() + () = (1)1

0

1

0

9

f R, : (2 + 1) =1

2 ()

7

1

3

0

10

f f(x) 0 [ , ] :

() > 0 () < 0

11

f , g g f fg,

12

f , g , h h (g f ) , (h g) f h (g f )= (h g) f

-

- 16 -

14

f : R 1 1 , x1 , x2 A : f (x1 ) = f (x2 ) , x1 = x2

15

f 1-1 y f (x) = y x.

16

f 1-1

.

17

, 1 -1.

18

f .

19

f : R. 1 ( ) , f f x x x A

20

f : R. 1( ) = , ( ) f f y y y f A

21

f f -1 y = x.

22

f f - 1 y = x.

23

xf(x) = 10

g (x) = logx.

24

1 -1 .

25

f 1-1 , f (x ) = 0 .

-

- 17 -

26

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

27

0 0

( , ) ( , )x x l .

: 0 0

x x x x

limf(x) l lim(f(x) l) 0

.

28

0

lim ( ) 0

x x

f x , f (x ) > 0 x0 .

29

0

lim ( ) 0

x x

f x , f (x ) < 0 x0 .

30

f (x ) < 0 x0 0x x

lim f(x) 0

.

31

f x0 0

lim ( ) 0

x x

f x ,

0

lim ( ) 0

x x

f x .

32

f g x0 :

f (x )g(x ) x0 , 0 0

lim ( ) lim ( )

x x x x

f x g x .

33

0

lim( ( ) ( ))

x x

f x g x ,

0

lim ( )x x

f x 0x x

lim g(x)

.

34

:. 0

1lim 1

x

x

x

35

: lim 1

x

x

x

36

0

lim ( )

x x

f x , f (x )>0 0 .

37

0

lim ( )

x x

f x , f (x )

-

- 18 -

38

0x x

lim f(x)

, 0

lim ( )

x x

f x

39

0

lim ( )

x x

f x , 0

lim ( )

x x

f x

40

0

lim ( )

x x

f x , 0

1lim 0

( )

x x f x

41

0

lim ( ) 0

x x

f x f (x )>0 x0 , 0

1lim

( )

x x f x

42

0

lim ( ) 0

x x

f x f (x ) 1 : lim

x

47

11 1 0 (x)= ... , 0 : lim ( ) lim

x x

x x x a x x

48

0lim ln

xx

49

0

1lim ln

x x

50

x 0

7xlim

x

= 7.

-

- 19 -

51

f () f .

52

f () f .

53

f

( ) 0f x x f ()>0 .

54

f f

.

55

f [ , ] [ m , M ] m .

56

f [ , ] f () f () > 0 f ( , ) .

57

f [ , ]

x0 ( , ) f (x0 ) = 0, f () f () < 0.

58

f x0 , x0 .

59

f x0 g f (x0 ) , g f

x0 .

60

f x0

g x0 , g f

x0 .

61

f f .

62

f x0 , x0 .

-

- 20 -

63

f x0 , x0 .

64

f Bolzano , f .

65

f x0 , f x0 .

66

f , g x0 ,

f g x0

0 0 0

( ) ( ) ( ) ( ) f g x f x g x .

67

f , g x0

0

( ) 0g x , f

g x0

:

0 0 0 0

0

0

( ) ( ) ( ) ( )( )

( )

f x g x f x g xfx

g g x .

68

0x 1

ln xx

.

69

: 1

(7 ) 7

x xx , xR.

70

f R , [ , ] , f Rol le .

71

f [0,1] ,

fC ,

0, (0) , 1, (1)f f .

72

f , f

.

-

- 21 -

73

2 .

74

f ( , ) x0 , f . f (x ) ( , x0 ) (x0 , ) , f (x0 ) f ( , ) .

75

f , .

f , f (x ) < 0 x .

76

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

77

f x

. f ( ) 0 f x

x .

78

f

. f (x) 0 x ,

f .

79

f , g .

f , g f (x) g (x)

x , f (x ) = g(x )

x.

80

, f 0,

f .

81

f [ , ] x0 [ , ] f . f (x0 ) = 0.

-

- 22 -

82

f ( , ) , x0 , f . f (x0 )>0 ( , x0 ) f (x0 )0 x , f .

84

f f (x ) > 0 x .

85

C f .

86

f , C f C f .

87

f ( , ) , 0 . f ( , x0 ) (x0 , ) , (x0 , f (x0 ) ) c f .

88

3

23 2f(x)dx f( ) f( )

89

5

5

2 2

17

7dx ln x

x

90

f , ,

( ) ( )( )a

f ff x dx

.

91

f [ , ] R ,

( ) ( ) f x dx f x dx

.

92

f [ , ] ,

-

- 23 -

( ) ( )( ) ( ) f ff x dx xf x dx

.

93

f , g R, :

( ) ( )( ) ( ) ( ) ( ) f x g xf x g x dx f x g x dx

.

94

f , ,

:

f(x)dx f(x)dx f(x)dx .

95

f [ , ] [ , ]

f (x ) 0 ( ) 0 f x dx

.

96

( ) 0 f x dx

, f (x ) 0

x [ , ] .

97

f [ , ] . G f [ , ] ,

( ) ( ) ( ) f t dt G G

.

98

f(x)g (x)dx f(x)g(x) f (x)g(x)dx ,

f , g [ , ] .

99

f , g , g [ , ]

, ( ) ( ) ( ) ( ) f x g x dx f x dx g x dx

.

100

( ) f x dx

xx xx.

-

- 24 -

-

- 25 -

.

-

- 26 -

-

- 27 -

-

- 28 -

-

- 29 -

..1. A1.A f ' x0

,

f (x0 , f (x0 ) ) .

2. , f '

x0 ,

.

3 .

.

. f x0 , f

x0 .

. f x0 , f

x0 .

. f x0 , f

x0 .

4 .

x0 .

. f (x )=3x 3 , x0=1

1. y=-2x+

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

2 2. y=

1 4

x+1

. f (x )=3 x , x 0=0

3. y=9x-6

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

4. y=-9x+5

5.

-

- 30 -

..2. 1. Fermat.

2. f ,

' 0f x .

f .

3 .

.

1. :f A

, f .

2. 0

lim 0x x

f x

, f x 0x

0

lim 0x x

f x

.

3. f 0f a f 0f x

,x a , f , . 4. f , g

' 'f x g x x ,

.

5. f [ , ] ,

: ( ) ( )f x dx f x dx

.

6. f ,g

,x , a

f x dx g x

.

..3. 1. ,f g .

,f g

' 'f x g x ,

c , x

: f x g x c

2. , 0, 1vf x x v

: .

3 . 0

x

F x f t dt , f

.

.

:

f x

x

f x g x

x

,a

f x g x

1' vf x v x

36 . .

-

- 31 -

.

.

. 10F

..4. 1. f 0x

. f

0x , :

0' 0f x .

2 . x

f ;

3 . f

, ;

4 .

() () ;

1. 0

limx x

f x l

00

lim

h

f x h l .

2. 0 1a lim 0xx

a

.

3. f , f

f a f .

4. f g

,a : ' 'a

f x g x dx f x g x dx f x g x

5. f x ,

f x x f 1-1 .

..5. 1. f

0x .

2.

)(, 00 xfxM f .

3 .

.

) f (x ) = ln(x2+1) [0 , +)

0F

4F

-

- 32 -

) axg )( 0x axgxx

)(lim0

( )y aim f y l

0

( ( )) .x xim f g x l

) f [, ] f () , .0)(' f

) f , .0)('' xf

) f [2,5] 0)( xf

[2,5] , .0)(

2

5

dxxf

..6. 1. f ,

, . :

f ,

f f

, f f ,

0 ,x a , 0f x .

2 . 0x x

f ;

3 .

Rolle .

4 .

() () .

1. : 0

1lim lnx x

2. :

xxim e .

3. : : f A R : g B R ,

f

g, .

4. f 0x

, 0x .

5. ' 'f x g x dx f x g x f x g x dx

', 'f g

, .

..7. 1. , f

' x0 ,

-

- 33 -

.

A2. f

x0A;

A3. f (x ) = x , >0

R f (x ) = x ln.

A4.

:

i . 1 -1

.

ii . f (x ) = e x+1 .

iii . 0

limx x

f (x)>0, f (x)>0 x0 .

iv. x , y

y = f (x ) , f x0 , o

y x x0

y = f (x0 ) .

v. f ,

x0 , f (x0 )0 ,

f .

..8. 1. f ,

.

. f (x)0 x ,

f .

. f (x)0 x ,

f ;

2 .

.

. f (x) =e1 - x

.

. f f (x) = -2x+2

1

x + 3, x

2,)

.

. f (x) = g (x ) + 3 x,

h(x)=f(x)-g(x ) .

-

- 34 -

..9. A1. f [, ] .

G f [,] , :

).()()( aGGdttf

3. f .

f x0A ;

2.

f -2,6 .

f

..10. A1. f(x )= , >0

R xR f (x ) = ln .

A2. f, .

f .

A3. ,

, , ,

.

. f (x )=logx, x>0 g(x)=10 x .

. f

x 0 A () f (x 0 ) , f (x ) f (x 0 ) xA

-2 1 3 6x

y

-

- 35 -

. f ,

1 -1 .

. 0

lim ( ) 0x x

f x

f (x )>0 x0 , : 0

1lim

( )x x f x .

. f x 0

.

..11. A1. f x 0

. f

x 0 ,

: f (x 0 ) = 0.

A2. f . y=x+

f +;

A3. ,

, , ,

.

) 1

0( ) (1) (0)f x dx f f , f [0 , 1]

) f:A 1 -1,

x 1 , x2A : x 1x2 ,

f (x1 ) f (x 2 )

) x 1= {x/x=0} : 21

x x

.

) :x

xlim 1

x .

) C C f f - 1

y=x

xOy xOy.

..12. A1. f . f (x ) > 0 x , f .

-

- 36 -

A2. f [, ] ; A3. f . f

x 0A ; A4. , , , , , , . . , . . f 1 -1, y f(x)=y x .

.

0x x

lim f(x)= , f (x )

-

- 37 -

3. Rolle .

4. , , , , , , . . - f , xx, f . . x>0 lnx x + 1> 0.

. 0

-

- 38 -

2. f , g

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

c :

f x g x c x .

3.

.

) f :A 1f ,

f .

) f 0x ,

.

) f

, 0x , 0f ' x 0 . ) f

. f '' x 0

x .

) f , f x 0 ,

.

) f

, , f

.

..17. A1. vf x x , v IN 0, 1 .

f v 1f ' x v x .

2 . f ,

,

.

0 0f x

0f x x 0x

0f x dx

0 ,x a 0 0f x

0f x dx

-

- 39 -

, 1 2 3I , I , I

.

3 .

.

1. x 0

xlim

x

2. x 0

1lim x

x

3. x 0lim ln x

4. xx

1lim

e

.

. 0

. 1

.

..18. A1. f , g x0 ,

f + g x0 :

( f + g ) (x0 ) = f (x0 ) + g (x0 ) .

2 . f .

f ;

3 .

.

3

10

I f x dx

3

20

'I f x dx

3

30

''I f x dx

-

- 40 -

1. f : R 1 - 1

1 2x ,x 1 2x x 1 2f x f x .

2. 0 0x x x x

lim f x lim g x

f x g x 0x .

3. f ,

, .

4. f ,

, 0f ' x 0 .

5.

f x dx 0 f

, .

..19. 1. f

0x . f

0x ,

: 0f ' x 0 .

2.

f ;

3.

.

. f [2 , 5]

.

.

.

. 0x x

lim f x

f x 0 x

0x .

. f

,

f()0 x .

. 0f x dx

0f x

,x a .

..20. A.1 : ( ln|x| ) = 1

x.

.2 f

. f

0 ,x a 0 0f x 0f a f

0 ,x a

, 0f x ,x a

0x x

0

limx x

f x g x

0

limx x

f x

0

limx x

g x

-

- 41 -

;

3 . ,

.

. f f () 0 x R

. 1

ln( ) , ( ,0) x xx

-

- 42 -

..22. A1. f ' [, ] .

G f [, ] ,

f (t) dt G() G() .

2 . f .

f ;

3 . ,

.

. f [, ]

(, ] , f [, ] .

. , 1-1 ,

.

. f x0 0x x

lim f (x)

=0,

x x0

lim f(x) 0 .

. f R ,

f (x)dx xf (x) xf (x)dx , f [,] .

.

,

.

..23. 1 . , f

x0 , .

2 . y =

f + ;

-

- 43 -

3 . ,

.

. f ' (, ) ,

x0 , f

.

f (x ) > 0 (, x0 ) f (x) < 0 (x0 , ), f (x0 )

f .

. f

. f (x )>0

x , f .

. f ,

[ , ] ,

f (x)dx f (x) .

. f ,

f

.

. f x0

. f x0

f (x0 )=0, f

x0 .

..24. 1 . f .

F f , :

. G(x) = F(x) c ,c R

f

. G f

G(x) = F(x) c ,c R .

2. f

x 0 ;

-

- 44 -

3. ,

.

. 0x x

lim f (x) l

, 0x x

lim f (x)

0x x

lim f (x) l

. f , g x 0 ,

f g x 0 :

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

. f ,

. f (x)>0 x , f

.

. f , g ,

:

f(x) g (x) dx f(x) g(x) f (x) g(x) dx .

..25. A1. : 1 , 0,2

x xx

.

2 . ,

.

. f x 0

, .

. , ,

(x0 , f (x 0 ) ) , C f f,

x 0

= f (x 0 ) .

. f , g IR

f og gof ,

.

. C C f f 1

y = x

xOy xOy.

. f x 0 , 0 0

kk

x x x xlim f(x) lim f(x)

,

f (x ) 0 x 0 , k k 2.

-

- 45 -

3 . f

(, )

[, ] .

..26. 1 . : 1 * * , x x x R .

2 . y = x +

f -;

A3. ,

.

. f [, ] f () < 0

(, ) f ( ) = 0, f() > 0.

. 0x x

lim f(x) g(x)

0x x

lim f(x)

0x x

lim g(x)

. f f - 1

f y = x,

f - 1

.

. f R * .

f R *

f (x ) = 0 x R * ,

f R * .

. f

, x

x ,

.

..27. 1 . (x ) :

0

0lim ( ) ( )x x

P x P x

.

-

- 46 -

2 . f: A IR 1 -1;

A3. ,

.

. , f

0,

f .

. f (, )

x o . f (, x o )

(x o , ) , (x o f (x o ) )

f .

. f [ , ] IR,

:

f(x)dx f(x)dx .

. f , g fog gof,

fog gof.

. f (x ) = x, xR f ()=-.

..28. A1. f .

F f , :

:G(x)=F(x)+C, C R

f G f

: G(x)=F(x)+c, c R .

2.

.

.

f (x)dx = . . . . .

f (x) g(x) dx = . . . . .

f (x) g(x) dx = . . . .

, R f ,g [,] .

3 . :

. 1

x

0e x dx

.

24

1

3x dx

x

-

- 47 -

.

2

02x 3x dx

..29. A1. : ()= 1.

2 . .

3 . ,

.

. 0x x

lim f(x)

,0x x

lim f(x)

+ ,

0x x

f .

. f , g x g(x )0,

f

g x

:

o o o oo 2

o

f f(x )g (x ) f (x )g(x ) x

g g(x )

.

. x0 1

ln x x

.

. f:R 11,

y f(x)=y

x.

. f [,]. G

f [, ] ,

f(t)dt G() G() .

..30. A1. f ( , ) , x0 , f . :

f () ( , x0 )(0 , ) , f (x0 ) f ( , ) . 2. R.

-

- 48 -

;

3 .

,

, , .

1. f x 0 .

f (x )0 x . 0x x

lim f(x)

0x x

1lim

f(x) .

2. x 0, 2x 0

1lim

x .

3. f

x 0 , x 0 .

4. f(x) x = [0, +),

1f (x)

x x (0, +).

5. f , g .

f , g f (x ) = g (x)

x , c , x

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

..31. A.1 f

x0, .

.2 f ;

. ,

.

. f ()

f .

. f , g, g [, ] ,

f(x)g'(x)dx f(x)dx g'(x)dx .

-

- 49 -

. f

,

/

f(t)dt f() - f()

x.

. f

(, ) ,

(,) = x lim f x

=

x lim f x

.

. f x0

R, :

o ox x x x

lim k f(x) k lim f(x)

k R .

..32. 1. f x x , {0,1} .

f R

1f x x .

A2. N f

.

A3.

( ) ,

, () , .

1. f x 0 g

x 0 , gof x 0 .

2. f 1 -1,

( xx)

.

3. f x0

R 0x x

lim f x 0

,

f (x )

-

- 50 -

..33. 1. : f , g

x0

, f + g

x0

: ( f + g) (x0) = f (x

0) + g (x

0) .

2. f g ;

3 .

( ) ,

, () , .

1. f [, ]

x [, ] f (x ) 0

f(x)dx 0 .

2. f ,

xx, f .

3. f, g, h h (g f ) ,

(h g) f h (g f ) = (h g) f .

4.

2 .

5. > 1 xxlim 0

.

..34. A1.

.

A2. f

[, ] ;

3 . ,

, , , ,

.

-

- 51 -

. f:A IR 11,

f - 1

: 1f (f (x)) x , x A

1f (f (y)) y , y f (A) .

. f

f

.

. f

x .

f f(x) > 0

x .

. f IR

,

f ( x ) > 0 x.

. f ,,

f(x)dx f(x)dx f(x)dx .

..35. A1. [, ] .

G f [, ] ,

f(t)dt G() - G() .

2 . f .

3 . ,

, , ,

.

. 11,

.

. f ,

f

,

.

.

f(x)dx

xx

-

- 52 -

xx.

. f

, x

x, .

.

(, x 0 ) (x0 , ) .

: 0 0x x x x

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

.

..36. 1. f .

f x

f (x ) = 0 , f

.

2 . f x0

;

3 . ,

, , ,

.

. f 1 -1,

f .

. f

() x0

A, f (x)f(x0) xA.

. x 0

x 1lim 1

x

.

. f

.

. f [, ]

f (x )

-

- 53 -

..37. A1. f .

F f , :

G(x) F(x) c, c

f

G f

G(x) F(x) c, c .

A2. .

A3. f

.

f

;

4. ,

, , ,

.

) 0

limx

x=0

x

) f

. f ,

.

) f

(, ) ,

(,), x

A lim f (x)

x

B lim f (x)

) (x)= x, x

) 0xx

lim f (x) 0

, f (x) 0 x 0

..38. 1. , f

x 0 , .

2. f (x0 , f (x0 ) ) C f . C f ;

-

- 54 -

3. (5) , . .,

, , ,

.

. f

C f

.

. f

c, : cf (x) f (x) , x.

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

. f

[, ] [m, M],

m .

. 0x x

lim f (x)

, 0x x

1lim 0

f (x) .

..39. 1. f,

[, ] . f [, ]

f ()f() , f () f ()

x0

(, ) f (x0)= .

2.

.

3. ,

, , ,

.

) : x

xlim 1

x .

) f f

C f , xx,

, xx,

C f , xx.

-

- 55 -

) f , g x o , f (x )g(x)

x o , : 0 0x x x x

lim f (x) lim g(x)

.

) f , g x o g(x o )0,

f

g x o

:

0 0 0 0

0

0

x x x xx

x

2

f g f gf

g g.

) P(x) , Q(x) .

P(x)

Q(x), P(x)

,

.

..40. A1.

f f - 1 y = x.

A2. Bolzano

.

A3. f A. f

x o () , f (x o ) ;

A4. ,

, , ,

.

) f

. f

, f (x ) > 0 .

) ox x

lim f (x)

, ox x

1lim 0

f (x)

) f () ,

.

-

- 56 -

) f

, , , :

f (x)dx f (x)dx f (x)dx

) f

. f

,

.

-

- 57 -

-

- 58 -

-

- 59 -

..1. f [0,] ,

0f(x)dx 2 F f .

1. F(0) - F () .

2. (0,) f ( ) = .

3. 0

( ) 1x

f t dt x

(0 , ) .

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

-

- 60 -

2. f

C f (0 , f (0) ) .

B3. f (x )+ 2x 2 xR.

B4. : ln ( ) ( ) 1f x f x .

..7. f

[0,4] , f (0) = 5 f (4) = 1.

1. f .

2. f (x ) = ,

[1 , 5] .

3. (0 , 4) :

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

6

f f f

f .

..8. f ,g (0,+ )

x > 0 : g xf x e f xg x e .

f (1) = g(1) = 0:

1. f , g .

2. h (x) = e - f ( x ) x , x>0

f .

3. : ( )

limx

f x x

x

,

0

( )lim

x

f x x

x

.

..9. f R , : 2016

40

0f (x) [f(x)] dx = 0 .

1. f (x ) = 0 (0 , 2016).

2. [0 , 1008 ]

f ( ) = f (1008 + ) .

..10.

f :R R (2 , 5) , ( -1 , 3).

1. f .

2. f .

3. : f(2x 1) f(5) f ( f (x ) ) = f (5) . .

4. : f - 1 (5) , f - 1 (3) .

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

6. C f (9 , 9)

f - 1 .

-

- 61 -

..11. : 31f(x) x x2

.

1. f

1f .

2. : 1 .( ) 64 f x 3. f - 1 ,

: 1f (1).

B4. :

1

6

( ) 64lim

6x

f x

x

.

..12. f f(x y) f(x) f(y) 2xy

x , y R

x 0

f (x)4

xlim

.

1. f 0.

2. f .

3. f .

..13. : 4

f(x) 2x , x 0.x

1. ( ) f ,

xx x = , x = +1, >0,

( ) 2 1 4ln( 1) 4ln .

2. ()

.

3. : lim ( )

.

..14. . :

i ) 1 0

lim tx

x tdt

e

ii ) 0

lim tx

x tdt

e

. :

i ) F(x)=2

1

2

x x t

dtt

.

ii ) G(x) =2 3

2

3 4

xdtt

.

iii ) 12

3x H(x) dt1

lnt.

-

- 62 -

..15. f g

R : f (x ) g (x) = x xR.

, g < 0 < :

1. f (x ) = 0 ( , ) .

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

..16. . f :RR

f (x)

f (x)e dx 0, , R < .

:

f (x) = 0 (,) .

. f R

: 3( ) ( ) 1 xf x f x e xR .

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

. f R

() f

: ( , f () ) , ( , f () ) ( , f () ) .

x0R f (x0 ) = 0.

..17. . : [ , ] ,f

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

i ) f (x )=2x

( , ) . ii ) 1 , 2 ( , ) :

f (1 ) f (2 )=4.

. : [0,1] (0, )f

f (0)=1, f (1)=2 : =2

1

0

f (x)dx

f (x) f(x)

.

..18. . : e

f(x) lnx xx

.

i ) f .

ii ) : 1f (x) x .

B. f :[,] R f (x ) > 0

f (x)

2016f(x)f(x)

.

-

- 63 -

f () = 2016f () I =

f (x )dx .

..19. :21( )

2

xf x e x .

B1. ( , f ( ) ) ,

(1 , 2) , C f , C f ( , f ( ) ) ,

: x + 2y = 1.

B2. f R.

B3. R g ,

: ( ) ( ) ( )f x g x f x xR.

g(0)=0 g.

..20. f : R R R 3f (x) f (x) x 5 0

x R .

B1. f

.

B2. : 1lim ( )xx

e f x

.

B3. f R

f (x ) = 0.

4. C f

(3 , f (3)) .

B5. (3 , 5) , : 1

( )2

f .

..21. : f (x ) = x + xe - x . 1.

fC

(0, f (0))

2x y + 7 = 0.

2. = 1, f

y = x fC .

3. () fC

, y = x x = 0 , x = > 0 , = 1.

4. :

E()lim .

..22. : f (x ) = x e- x

+ x . 1. R , c f (0 , f (0) )

( ) : 2y - 4x - 5 = 0 .

2. f

.

3.

-

- 64 -

c f .

4. f .

..23. f [1 , 3] .

1. f (1) = f (3) , x1 , x2

1

-

- 65 -

f x f xf(x)

1 1

2.

..28. f R ,

f (x )2 xR.

2

25

0( ) 5 1 ( ) , .

x xg x x x f t dt x R

:

. g(-3) g(0) < 0.

. g(x ) = 0 ( -3 , 0) .

4 1997

..29. h: [1 , + ) R

: h(x) (x ) F(x)1999 1 1, F

( )

( )h x

f xx

h(1)=0.

B1. h.

B2. h [1 , + ) .

..30. f R .

2 2 2 4

1

0( ) ( ) 2 ( )

I x f t xt f t x t dt , R

x0 = 2

1

05 ( ) t f t dt.

1 2000

..31. f f (x ) = x 2 lnx , x>0.

1 f .

2. 1

( )

x

f x x x , x > 0.

3.

f , xx 1

xe

x = e .

..32. x

x

e 1f x , x R

e 1

.

. f

1f .

-

- 66 -

. 1f (x ) = 0

.

. 1

21

2

f x dx .

2 2002

..33. f f (x )=x 2 lnx .

. f,

.

. f

.

. f .

2 2004

..34. f: IR IR f (x ) = 2 x + m x 4 x 5 x ,

m IR , m > 0.

. m f (x ) 0 x IR .

. m = 10,

f, xx

x = 0 x = 1.

2 2004

..35. f (x ) =2+(x -2)2

x 2.

. f 1 -1.

. f - 1

f

.

. i .

f f - 1

y = x .

i i .

f f - 1 .

2 2006

-

- 67 -

..36.

x

x 1

1 ef(x)

1 e

, x IR .

. f IR .

. 1

dxf(x)

.

. x

-

- 68 -

..39. f, R . A

x0 xf(x )=x+2x, :

B1. f (0) .

B2. f (x)

-

- 69 -

: x5y+2010=0

B3. f 5

2(+).

..43. f :RR f (1) = 1

xR : e x f (x ) + e x f (x) + f (x) = 0

1. f

2. N 1

0( ) f x dx

..44. f :RR R

( )( ) 0

f x

xf x

e f (0)=1.

B1. N f f (x ) > 0

xR.

B2. A (0, 2 )

2

2

2 2 2ln(1 )

2

ee

e

B3. : 20

2 2

2

xdx

x e.

..45. f 0,4

, F f

2 2(0)

4 32 2F F

.

B1. 0,4

,

f ( ) = .

B2. 2

( )lim

( )

x

x f x

x .

-

- 70 -

..46. : x

2

f (x) x xtdt 14 , , ,x 0.

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

M(2 , 6) .

2. , 1.

)

.

)

.

) .

..47. f : [0 , 5]R f (0) = 0 ,

f (3) = f (5) = 6.

B1. x0 (0 , 5) , : 5 f (x0 ) 6 = 0.

2. x1 (0 , 5) , : 5f (x1 ) - 2 = 0.

..48. ln x

f x , 0x

.

1 . f

(1, f 1 ) x y 0 , .

B2. = 1:

. f .

. .

. : 1

1

8 .

2 2003

..49. f , g

x .

, 0

0.

1 . i ) L.

ii )

f g . 2 . g .

3 . : x .

2 2004

' ' 1, ' 1f x g x f x

2lim

2x

g xL

f x x

4f x g x x

-

- 71 -

..50. f x 2 x ln x 2 , x 0 .

1. : ln x

f ' x , x 0x

.

2. x 0lim f ' x

.

3. f

.

4.

ln x

g xx

,

. 2 2005

..51. 2 ,xf x x a e x . f

:

1. : 2 .

2. f .

3. :

i . xlim f x ii .

xlim f x .

4. 2007f x

. 2 2007

..52. f

, 0

1 1, 0

x xf x

x x

, .

1 . , f .

2 . , f

0 0x .

3 . f 1-1.

4 . 1 2 , 2 f x dx

.

2 2008

..53. :f : 3 24 12 1f x x x x ,

x R R 0 1x .

1 . i . =1.

ii . f .

2 . :

3xf x

imf x

.

'x x

1x

e

2x e

2 2y x

0, 0f

-

- 72 -

3 . i . f

0, 1 . ii .

f 'x x . 2 2011

..54. f (x ) = e x - 2 g(x) = lnx+2.

B1. f g g f

.

B2. f f - 1 .

B3. : 2 ln 2xe x ,

(e - 2 , 2) .

B4. :

( ) ( )

lim lim 0.( ) ( )x x

f x g x

g f x f g x

2 2012

..55. : f (x ) = - e 3 x x3 + 1.

B1. f

.

2. :

0

( ) 1lim

( )x

f x

f x

.

B3. ) f

.

) 3 3 2015 1

xe xe , .

4. g (0 , +)R :

33 ( ) 3 3 6( ) ln 2g xe g x x e x , x>0,

g g(x)=lnx+2 .

2016

..56. R f

:

f ()=e , f ()=e , f ()=e , ,,R

-

- 73 -

1. f (x) f (x) = e 2 x 2 ( , ) .

2. x0 ( , ) , :

( f (x0 ) )2 + f (x0 ) f(x0 ) = 2e 2 x 0 .

University of Bristol,

..57. f [0 , 1] ,

f (0)=0, f (1)=1 0 f (x )1 x [0 , 1] .

:

1. x0 (0 , 1) , f (x0 ) = 1 x0 .

B2. , (0 , 1) , f () f ()=1.

,

..58. f R , :

f (0)=0 (x2 1) ( f (x) x3 ) 0 xR.

:

1. C f xx (0,0) .

2. f (1)=1 f ( -1)=-1

B3. ( -1 , 1) : f ( )=1.

University of Oxford,

..59. P(x) :

2 3( )

( )

x xf x

P x

, R.

P(x) R ,

f :

lim ( ) lim ( ) 1x x

f x f x

C f x=1 x=-2

f x0=-1.

University of New York

..60. >0 : 1

lim 1( )

x

xdt

t t

.

University of Oxford,

-

- 74 -

..61. 1

0

xx e dx * .

1. 1 .

2. : 11

( 1)e

, * .

3. :

) 1

3

0

xx e dx ) 1

01 1 xx x x e dx

BAC. ,

..62. :

4

( ) ln1

x

x

e ef x

e

.

B1. : f (4-x)=4-f(x ) , xR.

2. : 4

0( )f x dx .

BAC. ,

..63. f [0 , 1]

f (x)1, x [0 , 1] 1/2

0( ) 0f x dx .

:

1. 1

00

2

xf dx

2. ( )2 2

x xf x f

B3. 1

0

1( )

4f x dx

,

..64. 2 2

( )( )

txf t

t x

, tR x>0.

:

1. 2

1( )f t

x tR x>0.

-

- 75 -

2. 0lim ( ) 0x x f t dt

.

BAC. ,

..65. R f

:

1

0( ) ( ) 0f x f x dx

12

0( ) ( ) 18f x f x dx .

:

14

0( ) ( ) .f x f x dx

. . . 2004, Harvard University

-

- 76 -

-

- 77 -

-

- 78 -

-

- 79 -

..1. : ( 1) 6( )

xf x

x

x( -1 , +)

, R , y = 2 x = -1.

1. f : 2x 6

x 1f(x) , x 1

.

2. G(x ) G (x) = f (x ) , x >-1, (0,2) . 3.

G(x)h(x)

x 1

, x >-1.

..2. g:RR ( )1

x

x

e xg x

e

.

R f

: ( ) ( ) ( ) , f x g x g x x R f (0)=0.

1. f . 2. N f 1 : [0,] [0,] . 3. C f , C f - 1 x = 0, x = = 4..

..3. f [ 0, ] . :

1.

0 0

1f(x)dx f(x) f(-x) dx

2 .

2. x

( - x )x 0

2011 dx =

22011 2011 .

..4. F f :RR

xR : 2 2 22 ( ) ( ), F x F x 0.

: 1. F(0) = F(1) = . 2. f (x ) = 0 R.

..5. : F(x)=1

4x2 (2lnx-3)-x(lnx-2)

x > 0. 1. F F(x ) = 0 (0 , + ) .

2. :

2xf(x) x ln x

2

23xg(x) 2x

4

: 2004

( )

( )

1

1f x

g xdt

t >0 x >0.

..6. A. 'f R F

f R.

-

- 80 -

N R :

5 ( ).( ) ( 5) ( ) f xg x F x F x

B. f : R R f (x+f(y)) = f (x+y )+2 x, y R .

:

) f (x ) = x + f (0) .

) f (x ) = x + 2 xR.

..7. f :R *R 1 -1 f f (x) f(x)

xR * , 0, : 1. f f (x) x 0.

2. f .

3. f .

4. f .

5. 1f .

..8. :f (0, ) R x

f(x) f(y) fy

x, y > 0. f (x ) = 0 , :

1. f 1-1.

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

3. f (x) 0 x > 1 f

.

..9. f : RR , :

f ( lnx) = x x>0 f (0) = 0. 1. f . 2. : f (x ) > x 2 xR.

3. : 32

1

2f(x)dx .

..10. : f (x) 2x x,x [0,].

1. 1f

.

2. 1f (x) x.

3. =2 1

0f (x)dx .

..11. 1. , :

1 1

0 0

x (1 x) dx x (1 x) dx .

-

- 81 -

2. 2004f (x) x(1 x) , 2003g(x) x(1 x)

f , g.

..12. 1. : xx , x 0e x , x 0f (x) . )

0x 0 .

)

f x=0 , x=

2.

2. )

:

)

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

.

) f .

) : x

f (x)x

lim

.

..13. f 3f x ln x 1 x x e , x>-1.

1. f

f - 1 .

2. f - 1 (x ) = 0.

3. 1f ,

1f e .

..14. f

f : (0 , + ) (0 , + ) f (1) = 1 :

1 1xf

x f (x)

x > 0.

1. :f (x) 1 f (x)

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

2. f .

..15. f R *

:

1

( ) - ( ) xxf x f x e , xR * f (1) = e , f ( -1)=1/e.

1. f .

2. : 2

1/

31/.

( )

e

edx

f x

x

-

- 82 -

..16. :f R R 2 2(f f )(x) x (2 1)x x R , .

f 'f () 1

1. f ().

2. C f

( , f ()) .

..17. f 2

lnxf(x)

x .

1. C f .

2. .

3. , lnx

g(x)x

f .

4.

E()lim

()

C f x = 1 , x = >1 y=0.

..18. f R , f ( 3 ) (x ) > 0

xR. 1. C f . 2. (x1 , f (x1 ) ) (x2 , f (x2 ) ) C f .

..19. f R :

f2 (x ) 4 e x f (x ) = 1 xR f (0) = 2 - 5 . 1. f . 2. f .

3. : limx

f x .

..20. :f

1

2 1 7lim 10

1x

f x

x

.

1. :

) 3 7f

) 3 5f

2. f

3, 3f . ) 5 8y x . ) , 3,

. 2 m/sec ,

-

- 83 -

. ... 2008

..21. . f :RR f (x ) 0 xR. f (5) + f (6) + f (7) = 0 , f . . f R

f (1) = f (7) f (x7 ) f (7x ) xR. f (x ) = 0 (0 , 7) .

..22. . f : R R F f R.

F (x) 0 F(x) = F(2 x ) xR , f (x ) = 0 . . f : R R , : f (1 x ) + 2 = x f (x ) , x R. 1. f R .

2. : 21

( ) ln 1 e

f x dx e e .

..23.

2x x 1f (x)

x 1

.

1 . ()

C f , C f + x = 2 x = >2.

2 . : E( )lim

.

3 . 3 , t = 4.

..24. f 0,fD R.

x f x

f xx

0x ( ) : y = 2x e

Cf 0 0 x , y , : 1. .

2. f .

3. ln f x x x , f . 4. f .

5. 1

ln 02

xx

0,x .

-

- 84 -

..25. f f (x ) = x2 ( + )x + , < . C f xx. 1. . 2. 1 2 C f xx ,

: 1

2

3

2

E

E.

..26. f [0 , 4]

2 2

f 0 7 f 4 7 0 .

1 . f . 2 . [0 , 4] , :

1 1 12 3 4

2 3 4

9

f f f

f .

3 . f (x ) 0 x [0 , 4]

: 7 2xlim f(3) 1 x 2x 1

.

..27. f : RR , R

: x 0

xf(x) 1988xlim 18

x

2006x 7 x

xlim f(x) lim

x .

1. f (0) . 2. f ( -7) . 3. f y = -x ( -7 , 0) .

..28. f [1 , 3] .

1 . 1

3

2lim

x

f x

f x f (1) .

2 .

1

ln3

e xf dx

x , :

9

xf x

(1 , 3) .

3 . : 3 4 f x 1 , 2x :

1 2 2 f .

..29. . f [1 , 7]

: f (2) < f (1) < f (7) < f (5) .

(1 , 7) , f () = 0.

. f (x ) = xe - x , xR .

-

- 85 -

. f .

. : 2/ x

1/2 22 e xe dx e

.

4 1993

..30. f : R R ,

: f (x)

1f (x)

e 1

xR f (0) = 0.

1 . : x

xf (x) f (x)2

x >0.

2 . f , xx x = 0

x = 2 : > f (2) .

..31. f (0 , + )

e f (1) = 1

2f x

f xx

x > 0.

1 . (x) = f (x ) e 1 / x

x > 0.

2 . f .

3 . N

3

f xh x

x , xx

x = 1 x = 2.

..32. f , g R

:

) f (x1 + x2 ) = f (x1 ) f (x2 ) x1 , x2 R.

) f (x ) = 1 + x g(x ) xR.

) x 0

g x 1lim

.

1 . f R. 2 . f (x) = e x . f ) ) . New York University

..33. g(x) = x lnx.

1. : x

g(x)lim 0

.

2. :e lnx

I dxg(x)

2

1.

3. : g (x ) x x > 0.

4. : 1

ln 0xxe

x > 0.

-

- 86 -

..34. f :RR .

f f (3) = f (2) :

1. : f (x+1) = f (x ) .

2. : f (3x+1) > f (3x) .

3. : f (x6 + 2) = f (x6 + 1) .

4. f .

..35. 1 . f x

2 2

ef x

x

, >1

.

2 . x0 :

2x xe 1

, > 1.

.

..36. f 2 .1f x ln x 2x 1 , 0 , 22

1. f .

2. f = .

.

..37. . : x 1f(x) 2 ln2 g(x) ln 2x .

, (1,ln2) (2,ln4).

. f [ -,] : f (x )1

x ( -,) >0.

f()= f ( -)=- f (0)=0.

..38. f : RR :

f 2 (x ) +2f (x )x = x2 + 2x xR f (0) = 1.

1. :

g (x) = f (x ) + x , xR .

2. f .

3. 0

( ) 1lim

x

f x

x lim ( )

xf x .

..39. f [2,3] , (2,3)

f (x) 0 x (2,3) . :

1. f (2) f (3)

-

- 87 -

2. (2,3) :

5f ( ) = 2 f (2) + 3f (3) .

3. 1 ,2 (2,3) : f ( 1 ) f ( 2 ) > 0.

..40. R f , :

f (x )0 R

f(x) dx 0 , , R.

1. : = .

2. R

e

t

t e dt

22010

10 .

..41. f :R R

f (x) = -4x3e f ( x ) xR f (0) = -1.

1. N f .

2. N f .

3. : 20

1

1

my

dxx

ym

f .

..42. . f

[1 , +) f (1) = 1 1 < f () < 2008

2007

x > 1. : < f (x ) < 2008 1

2007

x x > 1.

. f :RR ,

:

f (x ) > 0 xR R.

f .

..43. f : RR :

(x2 + x +1)f (x) = e x (2x+1)f (x) xR f (0) = 1.

1. N f .

2. N f .

3. : 0 1

11( ) ( )

y ef e x y dx f x dx

e yM

f .

..44. f f (x) = 4e 2 x xR.

1. N C 1 , C2 f (x) = e 2 x + C1x +

C2 .

-

- 88 -

2. 0

( ) 2lim

x

f x ( )

2lim

x

f x

x, f .

3. e 2 x 2x 1 0 xR

4. e 2 x 2x = 2x2 + 1

5. N Cf - , ()

6. Cf , () ,

x =0 , x = -1

..45. :f

:

' 1 1f 0

(8 ) (3 )lim 5

5x

f x f x

x

.

fC 0, 0f 2,8 :

1. 0 2f . 2. 'f .

3. 1 1 3f .

4. 0,2 :

2 3 1 3f f

..46. :f R R

:

(1) ( 1)( )

2

f ff x x

, x R .

: 1. f (1) f ( -1) = 2

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

3. 1 2, 1,1 1 2 1 2( ) 2f f .

4. 1 2, 1,1x x 1 2x x 1 2

1 12

( ) ( )f x f x

..47. : 0,f

: 1 0f , 1 2f 2 1x f x x (1)

x 0, .

1. lng x f x x lnh x x x 0x .

2. f .

3. f 0, .

-

- 89 -

4. fC ,

.

5. 1

2015f x ,

(1,2) .

..48. :f

: 24 4xf x x f x e

44

f xf x x

.

1. f .

2. 24 xf x x e ,

limx

x

f x

.

3. fC

y y .

4. f

1fC 11, 1f .

5. 0 1,2x ,

fC 0 0,M x f x

1,5K .

..49. :f :

0

2lim 0x

f x

x

.

1. 0f .

2. 2

20limx

x f x

x

.

3. f :

2 2 1,x xf x e f x e x ,

. : ,x xf x e e x .

. : limx

f x

limx

f x

.

. f ,0

0, , f x k 2 2k .

-

- 90 -

..50. f [0,1]

f (x )>0 x(0,1) . A f (0)=2 f (1)=4,

:

1 . y=3 f '

x0 (0,1) .

2 . x1 (0,1) , f (x1 )=

1 2 3 4

5 5 5 5

4

f f f f

3 . x2 (0,1) ,

f (x2 , f (x2 ) )

y=2x+2000.

3 2000

..51. f ,

R , :

f3 (x ) + f2 (x ) + f (x ) = x3 2x2 + 6x 1 x R , ,

2 < 3.

1 . f .

2 . f .

3 . f (x ) = 0

(0,1) .

3 2001

..52.

x , x 1

f (x) x 1 1 e ln(x 1), x 1,2

R.

1 . x 1

1x1 elim

x 1

.

2 . R f

x o=1.

3 . =-1 (1,2) ,

f ( , f ( ) )

xx.

-

- 91 -

3 2001

..53. f , g R .

fog 1-1.

1 . g 1-1.

2 . :

g ( f (x ) + x3 - x ) = g(f (x ) + 2x -1)

.

3 2002

..54. f (x ) = x5+x3+x .

1 . f

f .

2 . f (e x ) f (1+x ) x IR.

3 .

f (0,0)

f 1f .

4 .

f 1 , x

x=3.

3 2003

..55. 2f(x) = x 1 - x .

1 . x lim f(x) 0

.

2 .

f , x - .

3 . 2 f (x) x 1 f(x) 0 .

4 . : 1

2 0

1 dx ln 2 1

x 1

.

3 2003

-

- 92 -

..56. g(x)=e x f (x ) , f

R f (0)=f(3

2) = 0.

1 . (0, 3

2)

f ()=f() .

2 . f (x )=2x 23x, :

() = 0

g(x)dx ,R.

3 . lim ()

.

3 2004

..57. f f (x) = e , > 0.

1 . f .

2 .

f , , y = ex.

.

3 . () ,

f ,

yy, () =e 2

2

.

4 .

2

()lim

2

.

3 2005

..58. f, IR

f (x)0 x IR .

1 . f 1 -1.

2 . C f f

(1,2005) ( -2,1) , 1 2f -2004 f(x 8) 2 .

3 . Cf,

Cf () :

1y x 2005

668 .

-

- 93 -

3 2005

..59. f(x ) = ex

e lnx, x > 0.

1 . f(x )

(1, +).

2 . f (x ) e x > 0.

3 .

2 2

2 2

x 2 x 2 4

2x 1 x 3

f(t)dt = f(t)dt f(t)dt

(0, +).

3 2007

..60. f(x ) = x 3 3x 22 IR

+ 2

, Z.

1 . f ,

.

2 . f(x ) = 0

.

3 . x1 , x2 x 3

f , (x 1 , f (x 1 ) ) ,

B(x2 , f (x 2 ) ) (x 3 , f (x 3 ) ) y = 2x 22.

4 .

f

y = 2x 2 2.

3 2007

..61. xln x, x 0

f (x)0, x 0

.

1 . f 0.

2 . f

.

3 .

xx e .

-

- 94 -

4 . f (x+1) > f (x+1)f(x) , x > 0 .

3 2008

..62. f (x )=x2 2lnx, x > 0.

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

2 .

f .

3 .

ln x , x 0

f(x)g(x)

k , x 0

i . k g .

ii . 1

k2

, g , ,

(0,e) .

3 2008

..63. xf (x) ln(x 1), x > -1 >0

1 .

1 . f (x) 1 x>-1 = e .

2 . = e ,

. f .

. f

1,0 0, .

. , 1 0 0 , , ,

f () 1 f () 10

x 1 x 2

(1, 2) .

3 2009

..64. f (x )=ln [ (+1)x2

+x+1] - ln(x+2) , x > 1

-1.

1 . ,

-

- 95 -

xlim f (x)

.

2 . = -1

. f

.

.

f

. f(x ) + 2

= 0

0 .

3 2009

..65. f(x )=2x+ln(x 2+1), x .

1. f .

2. :

2

2

4

3x 2 12 x 3x 2 ln

x 1

.

3. f

f

.

4.

1

1

xf (x)dx

.

3 2010

..66. f(x ) = (x 2)lnx + x 3, x > 0

1.

f .

2. f

(0,1] [1, +).

3. f(x ) = 0

.

4. x1 , x2 3 x 1

< x2 ,

(x1 , x2 ) ,

f ( ) f ( ) = 0

f (, f ( ) )

.

3 2010

-

- 96 -

..67. f : R R ,

R , f (0)= f (0)=0, :

xe f x f x 1 f x xf x x R.

1. : xf x ln e x x R.

2. f

.

3. f

.

4. xln e x x

0,2

.

3 2011

..68. y = x , x0.

(0, 1)

xy ,

.

t, t0 x (t)=16m/min.

1. ,

t, t0 : x(t )=16t.

2.

(4, 2) ,

-

- 97 -

, .

3.

.

4. t 0 1

0,4

d=()

.

xy.

3 2011

..69. f(x ) = (x - 1) nx - 1, x>0. 1. f 1=(0,1] 2=[1,+ ) . f .

2. 1 2013xx e , x>0

. 3. x1 , x2

x 1< x2

2, x 0 ( x1 , x2 ) , f ( x0 ) + f (x0 ) = 2012. 4. g(x) = f (x) + 1 x>0, xx x=e. 3 2012

..70. f:RR, :

xf (x )+1= e x , xR .

1. :

1 , 0

( )

1 , 0

xex

f x x

x

.

2. o f 1

.

3.

f (0,f (0) ) . ,

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

x .

4. 0

lim ln ln ( )x

x x f x

.

3 2012

-

- 98 -

..71. f ,g :R R , f

:

( f (x ) + x) ( f (x) + 1) = x , x R

f (0) = 1

g (x) = x 3 +

23

2

x 1

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

2.

f (g (x)) = 1.

3. x 0(0, 4

) ,

:o

0

x

4

f (t)dt = f (x0 4

) x0 .

3 2013

..72. f :R R :

2xf (x ) + x2 ( f (x) - 3) = - f (x ) xR

f (1)= 1

2

1.

3

2( )

1

xf x

x

, xR

f R.

2.

f 1.

3. :

3 2

2 25 1 8 8 1f x f x .

4. , , (0, 1) ,

: 3

2 3

0( ) 3 1f t dt f

. 3 2013

..73. h(x) = x - n(e x + 1) , x

1. h .

2. : ( 2 ( ) )

1

h h x e

ee

, x

-

- 99 -

3.

h + , - .

4. (x) = e x (h(x) + n2), x

(x) , xx x = 1.

3 2014

..74. :

ln

, 0( )

0 , 0

x

xe xf x

x

.

1. f x 0 = 0.

2. f .

3. i ) , x > 0,

f (x ) = f (4) x4 = 4 x

ii ) N x 4 = 4 x , x > 0,

, x 1=2 x 2=4

4. , , (2, 4) ,

: 2

( ) ( ) ( ) 2 ( )f f t dt f f

.

3 2014

..75. f :RR xR

12 ( )

21 3 ( ) xtf t dt

f x e , R-{0} .

1. :

i . f f () = -2 f2 (x ) xR.

2 2

1 ( ) .

3.

f x x R

xii

2. :

0( )tf t dt

.

3. f .

4.

, f x = ,

:

1 1

4 3E

a a

.

3 2005

-

- 100 -

..76. .

1.

f 0, 0f . 2. f

.

3.

f , 0, 0f 1x a .

i . : .

ii . .

3 2006

..77. f 1xef x e , x R .

1 . i ) .

ii ) 1'' 1 xxx ef x e e , f

.

2 .

f .

3 . f .

4 .

, .

3 2008

..78. : 2 2lnf x x x . 1 .

.

2 . f .

3. ln

2

x xg x

x

0 0x : 0g x g x

0x .

4 . 2x : 2 2 1 4f x f x f x . 3 2009

..79. : (0, )f ,

0x ( )

1 ( )

1f xx

x f xe

0)1( f .

) ( ) x

g x e x 1 -1.

) xxf ln)( x>0.

1xf x e a x 1

2

12

a aE a e a

alim E a

'f x ' , 'x x y y1

ln2

x

-

- 101 -

) ( ) 1

( )f x

h xx

.

) 0 , .2

x xx x

xe e

) h 21,xx

012 xx : 2 1

2 1

5

h(x ) h(x ) 1

x x 2e

.

3 2010

..80. : f R R :

1f x f x , x R . 1 . :

i . 2 1

2 2f

0 1 1f f

ii . 0 0, 1x , : 0 0 1f x x

2 . , , f 1

22

f x x ,

x R .

i . 2

'2

f

fC 2

2.

ii . :

0

1

x

f f xim

x

.

3 2011

..81. : xx

, x 0f(x) e 1

ln , x 0

.

1. (0 , + ) f

1

f (0)2

.

2. . f .

.

, .

3. x

0

1 12x dt

f(t) 1 2013

(0 , 1) .

3 2013

-

- 102 -

..82. f , g R f (1) = 1,

g(1)=0 :

() () = () 1 2() + 2 2 1 , .

1. :

() = g(x)+1.

2. ) g (1) .

) :

lim+

[( + 1) ( + 2

+ 1)] = 0.

3. () = ( 1)2 xR,

) f .

) R, (1 , )

h : () = (1 ) + 1.

3 2014

..83. f 0,2

,

: 2 2 2 1( ) 2 ( ) 1 , x 0, , f

2 6 2 6f x xf x x x

.

1. : f (x )=x-x , x 0,2

.

2.

( ) 1 , x 0,2

( )( )

1, x < 0

f x

g xx

x

R.

, g

.

3. =2, g(x) = 0

,02

.

4. =2, g 1 -1.

2016

..84. f :RR ,

xx f (0)>0.

f (x)

-

- 103 -

4. : 1 1

0 02 ( ) ( )xf x dx f x dx .

5. :

1

0(0) 2 ( ) ( )

01

f xf x dx f x

x x

,

(0 , 1) .

..85. f : (0 , +)R

: f (1)=0 ( )f xe x e x>0.

1. f .

2. ( )ln 0f xx e x>0.

3. : ( )

( )f x

g xx

, x>0.

M t = 0

( , g()) , (0,1) y = g(x) , x x=x(t) ,

y=y(t ) t0.

( t )

( t )=2 (t) ,

xx,

g ,

e.

4. 2 2

( ) 1 ( ) 2h x x g x , x>0.

h

.

5. : 2

1

1lim

( )

x

xxdt

g t .

..86. f ,g : [0 , 1]R .

1. f (x )g(x) : 1 1

0 0( ) ( )f x dx g x dx .

2. m f [0 , 1]

: 1

0( )m f x dx M

3. [0 , 1] , : 1

2

0

1( ) ( )

3x f x dx f .

-

- 104 -

-

- 105 -

-

- 106 -

-

- 107 -

..1. f :

f (x) > f (x ) x 0.

f (0) = f (0) = 0 h(x) = f (x) e - x , :

1 . h .

2 . 2( ) ( )x f x .

3 . x f (x ) > 0 x > 0.

4 . 7

0f(x)dx f(7).

..2. R f

: 2( ) = x 1 , xf x e xR f (0) = 3 .

1. : f ( 1998 ) < f ( 2016 ) .

2. (0 , 2016) , :

f 1821 2f 1940 3f 2016

f 6

.

3. 20

1xe x dx

f .

4. y = 4

f x0(0 , 1) .

5. : f ( lnx) = f ( -2x + 2) x > 0.

6. : 12xxf e x 2 02

, x > 0.

..3. f : [ , ] R

[ , ]. :

1. 1

f()

f()xf (x)dx f (x)dx .

2. 1( )

()( ) ( ) ( ) ()

f

ff x dx f x dx f f

3. 1 12

11

1 2ln

x

e

e

e ee dx dx

x e e.

4. f () = f () = :

1()

() ( ) = (2 f(x))dx

f

ff x dx x .

5. : xeI lnxdx e dx 21

01.

..4. f : R R

:x 2

2f (x) 7x3

x 2lim

f (5) =

x

x + 7x

xlim

.

-

- 108 -

1 . f .

2 . C f (2 ,

f (2) ) .

3 . f :

. f (x ) 5x + 3 0 xR.

. (2 , 5) f

.

..5. 1 . f

f (x ) = lnx

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

f g x = 1

x = , > 1.

3 . : E( )lim

.

4 . : x

2f (x) x 2x 1lim .

..6. :

x xx

g x e x , x f x e , x2

2

1 0 02

1 . g .

2 . x 0 g x 0 ,

;

3 . f

.

4 . 21 ex

0 1e dx lnx dx e .

5 . :21 x

0

ee dx

2 .

..7. : t

f(t)t

2 3

2, t [1,4] .

. : I f(t) dt. 4

1

. t

xx

g(x) f(t) e dtx

24

1

2

1 , >0.

i ) :

t

x x xe e e 2 2 2

1 4

t [1,4] >0.

ii ) : xlim g(x).

1 1999

-

- 109 -

..8. 1 . g(x) = x3 + x.

g ,

:g - 1 (x+1) = x+1.

2 . R f

: f3 (x ) + f (x ) x xR.

: 2

0

5( )

4 f x dx .

..9. f , g: (0 , + )R :

f (1) = g (1) = 0 f (x) + e g ( x ) = g (x) + e f ( x ) = 0 x>0.

1 . f g .

2 . h(x)=e - f ( x ) x, x>0

: f (x ) = - lnx.

3 . : e x > 1 f (1+x ) x>0.

..10. f : (0,+ )R xf (x ) f(x) = x

x >0 >1 .

1. f(1) = 0 , f .

2. f (x) - 1

e v x >0.

3. A Cf 2

3

e

.

4. Cf , xx

x= e , x = 3

1

e 3

.

..11. f : R R

( )lim

x

f x , ( )lim

x

f x f (x ) = ( )

2

1f x

e, xR ,

f (0) = 1.

1. f R

, R

x =0.

2. :

i ) f (x ) + e f ( x ) = 2x +1

ii ) H f f 1

iii ) O f f 1

.

3. Cf -

Cf + .

-

- 110 -

..12. f (x ) = e x ()

C f , x = ,

> 0.

1. () .

2. x o (0,) x = x o

.

3. 00

lima

x

a.

4. : 1

0

1ln

2

x

x ee dx .

..13. A. x > 0 , : 2 2ln

0 0

/

t txx e

e dt e dt .

B. g R ,

g(x )>0 2

( ) ( ) ( ) 0g x g x g x xR.

:

. g

g

.

. : 1 2 1 2( ) ( )2

x xg g x g x

x1 , x2R.

1 1997

..14.

f : [0,+ )R f (0) = f (0) = 0 ,

f (x) > f (x) x [0,+ ) .

N :

1. h : [0,+ )R h(x) = f (x)e - x

2. f 2 [0,+ ) .

3. 1 (1)

2 2

ff .

4. lim ( )

x

f x .

..15. f :

(x2 +1) f (x) + 4xf (x) + 2f (x ) = 0 xR.

1. g(x) = 2xf(x) + (x 2 +1) f (x )

R.

2. Cf (0,0)

y 2x +3 =0 f

3. f , ,

.

4.

5. Cf , xx

-

- 111 -

yy .

..16. f : (0 ,+) R f (1) = 1

e

x > 0 , 2

( ) ( ) x f x f x .

1. g(x) = f (x )

1xe

(0,+)

2. f .

3. f

C f .

4. 2

31

( )

f xdx

x.

..17. f (x ) = x3 + x + 1.

1 . f .

2 . f - 1 : f ( ) 1 1 . 3 . f (x ) = 0 R.

4 . : f (x)dx1

3

1.

5 . : 1

x 3

f (x) 1lim

x 3

.

..18. :

f (x ) =

3 1 , x 0

0 , x =0

x xx

|| <

1

.

1. f .

2. lim ( )x

f x .

3. f .

4. x o 1 1

,

f (x o ) = 0.

..19. ),1(:g,f

0 0 1f g , 1,x

2 22 2 0f x f x g x g x g x f x (1) , 0f x 0g x .

1. 0f x g x 1,x .

-

- 112 -

2. 1

1f x

x

.

3. f

.

4. f , xx

x , 1x 0 ,

lim

.

... 2008

..20. 1. :

f (x ) = 2 ( x-1) + lnx .

2. : ln x

g(x) x , > 02 x

.

3. :

ln

, 1, 4 .2

xx x x

x

4.

f , Ox

x = 1 , x = 4.

..21. : f (x ) = ln

, 0xe x

xx

.

1. f .

2.

x

x exa

e

x > 0

= e .

3. ) N 0<

-

- 113 -

..23. f : R R

: f (0) = f (0) = 1

(x2 +1)f (x )+4xf (x) + 2f(x) = e x , xR

1. f

2. f

3. : e x > x2 +1

4. : 3

3

1

( )(1 ) x

dxf x e

.

..24. f :RR ,

: x2 f (x) +4x f (x) +2f (x) > 0 xR.

N :

1. g : R R g(x) = x2 f (x )

R.

2. g .

3. f (x) > 0 xR .

..25. f(x) = , 0

2

0 , 0

x xx

x

.

1. N f

.Rol le 1

0 , 2001

.

2. :

x + xx = 0 2001 , 2

.

3. : 2

2

1

1

2dx

x x

.

..26. f : R R , 2 2

( ) ( )f x f y x y x , y R .

1. f R.

2. f 0.

3. : 1

0

1 1(0) ( ) (0)

3 3f f x dx f .

4. f R , :

2

( )lim .x

f x

x

-

- 114 -

..27. :f )(f

:

f .

f

f xf x e x x

:

1. To limx

f x

2. H f .

3. f