9o

1

. 1.

) f(x) = +

2

3 2x 4

x 6x 11x 6 ii) f(x) = 3 2 x 1 iii) f(x) = +32 log (x 2)

iv) f(x) = log x ( log4(3-x2)) v) =2

f 5x x(x) log 4 vi) f(x) =

24 x12

vii) f(x) 5 x 4= + viii) = 3 2f(x) x 1 x 2. R f(x) = ln ( x2+2x+9) = R

3. R

: )f (x)=+

24 - x(x - 1) x 1

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

+ 34 - x - x

) f (x) =2x - x

x - 2 - 1 + 1

3x - 8 - x

) f (x) = 5x - 3 - 1

) f (x) = log (x2 + x - 2) + log +x 33 - x

) f (x) = xe - 1 + 1 - lnx ) f (x) = x2x - 1

+ 1x - 1

, x [0, 2]

. 4. C f , : I) f(x) = x2-5x+6 ii) 3xf(x) e 1= iii) 5f(x) log (x 3) 1= iv) f(x) x 3 2= 5. f (x) = x2 - 3x + 2. ) f (1), f (0), f (-3), f (2) ) Cf ) f (t), f (xt), f (x + h), x, t, h R.

6. f (x) = x 1

lnx . ) f. ) f (x) = e x . )

7. ( ) ln( 1) f x k x= + + 2 1e 2 . : i) , R

9o

2

ii) fc 3 .

8. , :f g R R , : = + 2( ) ( ) 4f x g x x x R . fc gc .

9. , :f g R R : + + = 2 2( ) ( ) 1 2( ( ) ( ))f x g x f x g x , x R

10. x R f g

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

2 2

2

3 2

x +2x+3 -x -x-2

-x-1 2x

i) f x = x +2x+1 g x = x +2x+1

ii) f x = e g x = e

v) f x = ln e +1 g x = ln e +1

11. :f R R : 2 2( 2) f(3 x) 0f x + + = , x R . f .

12. ) + += =

2

2 3x 2 x 2xf(x) , g(x)x 1 x x

) + += =

x 3 x 3f(x) , g(x)x 2x 2

) = + = + f(x) x 4 x 5 , g(x) x 5 4 x

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

f1 (x) = 2x - 1

x - 1 f2(x)=

++

3

2x 1

x - x 1 f3 (x) = ( )2+x 1

f4 (x) = x (x 1 + 1) f5 (x) = lnex+1 f6 (x) = eln (x+1)

) R .

15. . f (x) = +x 1x - 1

, g(x)=2+ +2

22x 2x

(x - 1),R, x > 0.

) f, g ) f = g; . f=g .

.

9o

3

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

i) =

2

21( ) xf x

x x = + 1( ) 1g x

x

ii) =

2( ) ln

1xf x

x = ( ) 2ln ln(1 )g x x x

16.

+ += =

2

2x 3x 1, x [0,2] , x ( 2,1)3x 6f(x) g(x)2x 3 , x (2,6) x 1 x [1,13]

2f+3g, f.g , f 2g

17. f (x) =x,

+

>

2x 1, x 2 x 2

g (x) = ,

<

0, 0

g(x)1, 0

= >

: ) f g !!!!!!!!!!!!!!

.

19. = +f(x) x 3 = g(x) ln(x 2) . f g . g f . f f v. g g 20. f g, g f, f f, g g

21. f i) = + 2 ,(f g)(x) x x 3 g(x)=x-1

.

+ > += =+ +

2x 3, x 0x 1, x [1, )f(x) , g(x)2X 3. x ( ,1) 3x 1 x, x 0

9o

4

ii) = + + 2 4 ,(f g)(x) 3 x x g(x) = x2 iii) = + = (g f)(x) 5x 4, g(x) 7x 6

22. f [0, 2].

: ) f (x2) ) f (x - 4) ) f (lnx) ) 2f( 1 x ), ) f(3 x 2) ) f( 3 x)

x 3+

23. [ ]: 2,1f R

: = ( ) (2 3)g x xf , 2( ) ( )h x f x= ( ) (ln )q x f x=

24. : (0,1] Rf . ( ) ( 2) (ln )g x f x f x= +

25. 2( )2

xf xx

=

( ) 1g x x= .

: ,f g g f 1ff

26. x R + =3 3f(x) x ( ) =f f (x) x

27. f ,

+ = f(x) 12f 3, x Rx x

f.

28. f(x)

( ) ( ) ( )( ) ( )

( )

2 3 6 3

2 x 2

I) f 2x -1 = x -3x+2 II) f x = x -2x +1 x >0

III) f lnx = x - x+2 IV) f e -1 = x -3x+1

V) f(ln2x) = x+3 x >0

29. = 2 1( ( )) xf f x x R

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

30. f : R R, + +f( ) f() f() , . ) .f(x) 0 ) )f() f() f( ,R.

31. ( ) = + x ef g (x) e x = > .g(x) lnx 1, x 0 f 32. = f(x) g(x) x = 2g(x) x f(x), x R,

9o

5

= (f g)(x) (g f )(x) xf(x) 33. f : R R = + 2f(x 1) 2f(3 x) x 1, x R. ) = + +2f(x) 2f(2 x) x 2x 2 ) = +2f(2 x) 2f(x) x 6x 10. ) f

34. f 2f (x) - 3f ( 1x

) = x2, x0, f (2).

35. = +2f(f(x)) x x 1 .f(1) 36. = f(f(x)) 3x 2 = f(3x 2) 3f(x) 2, x R

37. : 3 1, 1( )2 , 1x x

f xx x

9o

6

) f gc c .

) ( )( ) , 0( )

f xh x xg x

= > hc

,x >0 .

44. f : R R (0) 0f =

=

( )( ) ,1x

f xg x x Re

. ( ) 0g x < 0x .

45. (1,5)A (5,-2) . : ) f . ) f f ( (e )) 2xf f < .

46. f : R R :

+ = + >( ln ) ( 1) ln 3, 0f x x f x x x . f , ( ) 2f x = ( 1) 2xf e < .

47. f : R R f ( ,0] f . :

) + = +( ) (5 ) (3 ) (7 )f x f x f x f x ) + = +5 3 7( ) ( ) ( ) ( )x x x xf e f e f e f e

48. f :R R . :

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

. f x > f 3 . f 2x+1 0 . f f 3x -1 < f f 2x+5

49. ( ) 1-x 1f x = e + -2 , x >0x

.

. f ( )0,+ . ( )0,+ : 1-xxe -2x +1>0 50. . ( ) ( )xf x = e +ln x+1 -1 . . . . : ( )2x 2e +ln x +1 >1 51. [ ) f : 0,+ R ( ) ( )2f x =3x+ln x +1 x>0 . . .

9o

7

. : ( ) ( )

2

2

4

x +1 +1ln > 3 x - x - 2

x +1

( )2, + 52. : 1lnx < -1

x

53. ( )g : 0, R+ (1,-2) , (2,-3) ( ) ( )f x = lnx - g x x > 0.

I. g . . . . ( )22lnx

+

g(x) 0> > 0.

.

59. , : [0,2)f g R

= + +( ) ( ) 2 f x g x [0,2) fc gc .

9o

8

60. , :f g R R + =2 2( ) ( ) 1f x g x

x R . fc gc 1=

=( ) 2 ( ) ( )h x f x g x 61. ) .

, . ) .

. -

62. . :

1. 2. 3. , .

63. :f R R . , f

,x R : ) 1 ( ) ( )( )( ) ( )

f x ff xf x f

++ =

+ ) ( ) ( ) ( )f x f x f + = +

64. f : R R

f (x + y) + f (x - y) = 2f (x) + f (y) x, y R. ) f . ) f . ) x R f ( x ) = f (x).

. 1-1

65. 1 1 :

1) 3f(x)= x -2 2) x+1f(x)= e -4 3) x -1f(x)=x -2

4)

2x -3 ,x 1f(x)=

lnx -1,x >1

5 )

2f(x)= x -6x+10, x >3

66. 1 1 :

9o

9

( ) ( ) ( ) ( )x

3x

2e -1) f x = 2x -4 ) f(x)= ) f x = 2- 4+3x ) f x = 3ln x+2 -1e +1

67. 1-1 ( ) ( ) ( )2 4 2i)f x = x -3 , ii) f x =5+ x -4 , ii) f x = x -5x +7 68. f ,g : R R :

f(x)>0 R g 1-1 15 3x-1f(f(x))= lnf(x)+g (e ) 1 1 .

69. . 1 1 1 1 .

70. f : R R : f (f (x)) x=

x f(x)g(x)= e +e , 1 1 . f 1 1 . g(f(x))= f(x) .

71. 2f(f(x))= x - x +1, x R . ) f(1)=1 ) g(x)= x 2 x + f(x)+1 , g 1-1

70. : R f R

( )( ) *, x += + f f x ax R : ) f 1-1 ) f R

) ( ) ( ) , x = f ax af x R ) ( ) , x

+=

f xf x R

a

71. : R f R : ( ) 5( ) ,f xf x e x+ = x R . :

f 1-1. H f .

72. f , g 1-1 g f 1-1

73. f: R g : f(A) R. g f

1 -1, : i) f 1 - 1 ii) g 1-1

9o

10

.

1-1 ( ):f A f A ( )1 :f f A A x A ( )y f A ( )f x y= ( )1 f y x =

1-1 ( f "1-1"f )

1 ffD R = 1 ffR D =

f , 1-1.

y x=

y x= , f .

( )

.. ( ) 1f xx

= ( ) 4f x x= . ( ) ( )1f x f x= . : ( )( )1f f y y = ( )( )1f f x x = ( )( ) ( )1 1f f x f x = :

1. 1-1

2. ( )f x y= f x 3. x f(A) f 1f 4. f 1f (x) 1-1

f -1(y)=4-y

y=3 x=f -1(3)=1

9o

11

.

74. ( ) xf x = e +1 . f 1-1 N f. . . f f-1

75. ( )x

x

e -1f x =e +1

. f . . f f-1

76. ( ) ( )xf x = 2+ln e +1 . f . . N f. . f f-1

77. 1( ) ln1xf x x

= +

) f . ) 1f . ) f 1f . ) : f 1-1 1f

.

78. . f 3x 2x(x)= 2- ln(3+2e ), f(x)= 3- 3-e ,

2015f(x)= 2x +5

79. f :[3, ) R+ 2f (x) x 6x 10= +

1. 2.

3. g :[1, ) R+ g(f (x)) x= x 2

80. : R f R ( ) 0f x > ln( ( )) ( ) xf x f x e=

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

9o

12

2. f 3. f 81. : R f R : 3f(x)-2f (x)+x -1= 0 x ) f ) 1f .

82. f(x+y)=f(x)+f(y) x,y R.

. f 0 f 1-1 .

. f -1(a+b)=f -1(a)+f -1(b) a,b R.

( f(R)=R)

( f(x-y))

83. f(xy)=f(x)+f(y) x R*+ . . f =1 *.

. f 1 1-1 .

. f 1-1 a,b f -1(a+b)=f -1(a)f -1(b).

f 1f : ) f ( )f x x= ( )1f x x = ( ) ( )1f x f x=

) f ( ) ( )1 y f x y f x= = 1f fx D D

84. :f ( )f R R= .

1. f 1f .

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

1 2(1 ( )) 1f f x x + + < .

85. A : A(3,2) B(5,9) f, : : 1 2f ( 2 f ( x 8x )) 2 + < f

: 1 2f ( 2 f ( x x )) 9+ + =

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

9o

13

86. ( ) 3 3, xxf x e x= + f ( )1 0f ( ) ( )1f x f x= ( )( )1 1 1f f x x + <

87. () =3+-1 .

= . ()= -1() . -1(3+2)>1

88. f : (3,2) (5,9). ) f ) -1 ) f(2+-1(2+))=9 ) f(-1(x2-8x)-2)

9o

14

) -1f y = x f R . 93. ( ) ( )f x = ln x -1 + x - 2 ) . ) ) -1f f f R . 94. ( ) 3f x x= . 95. ( ) 5f x = ln x+3x -2, x >0 f . fC 1fC

9

15

.

1. f . : )

+x 2lim

f (x) )

-x -1lim

f (x)

) 1

limx - +

f (x) ) 1

lim-x f (x)

) 1

limx +

f (x) ) 2

limx

f (x)

) 3

limx

f (x)

y

23

4

1

-2

-2

1-1 2 3 x

B. 0x x

f(x)lim

g(x) 0g(x ) 0

2. : )

++2x 2

3x 5limx 1

)

+ 45x 1lim ( (x 3) ) )

+

2x 3x 2 5

limx 6

v)

+ + + +

+

3

x 1

x 2 x 2x 1lim

x 3 1 v)

4 3 4lim 3 11

x xxx+

.

3. : i)

2

2x 1

x -1limx - 3x +2

ii) 1x

lim

31 3

1 x 1 x

iii)

2x 2

(x 2) 3 xlimx 4

iv)

+ +

2

3 3x x ( 1)x lim

x

4. 3 3

2 4x 1 x 1

x -1 x -3x+2lim ) lim2x +3x -5 x -4x+3

)

.

00

9

16

.

5. : i)

+

2

x 2x 4lim

3x 2 5x 2 ii)

+ +x 11 3x 2lim

3 5x 4 iii)

+ 3 3

x 01 x 1 xlim

x

6. : i)

3

4x 1x 1limx 1

ii)

+ +

3

x 17 x 3 xlim

x 1

iii)

x 3

f(x) 2 f(x)limf(x) 1

=x 3lim f(x) 1 iv)

1limx

2

2

x - 1 + x - 1x - 1

.

7. 2

2 3x 2 x 1

x +7 -3 x -1- 2x - x -1lim limx -4 x -1

a)

)

.

8. : )

2x 2

x 1 x 3lim

x 4 )

+

+ +

3

3x 1

x 3x 1 xlim

x 5x 4 2

)

+

2 2

x 3

x 9 x 6x 9limx 1 2

v)

+ =

2

2x 2 x 2,

f (x) 4 3 f(x) 5lim , lim f(x) 1

f (x) 1

9. : ) 3 3 1

lim 1 21

+

+

x xxx

)

+ +

+

2 2

x 1

x 1 x x 2lim

x 2 1

) 5 35 1 3 1

lim0

+ +

x x x xxx

.

10.

<

+= +

2

2

x 1, x 1x 1f(x) , x [ 1,1) (1,3)x 13x 1, x [3, )x 7

) lim f(x)

x 2 )

x 1lim f(x) )

lim f(x)

x 2 v)

lim f(x)

x 3

9

17

11. x 1limf (x)

( )

2

2

2

x -1 1

x -1

)

12.

( ) ((

2

2

x + x + x 0,2

f x = x +1 x 2,3

2x - 3 x 3,4

=2 =3

13. 2 2 2x f (x) 4f(x) 2x 5x 2 x 2

lim f(x)

14.

2x 0

5f(x) 3x 6 x , x R lim f(x) 15. f : R R + + +3 2x 1 f(x) x x x ,

i)

x 1limf(x) )

x 1

f(x) 1limx 1

)

+

2

x 1

f (x) 1limx 3 2

16. + + + 3 2x 1 f(x) kx x x, x R f

(1,1)

x 1

f(x) f(1)limx 1

17. + 23 x 5 9 (2x 4)f(x) 4(x 2x)

x 2 x 2 3

f(x) 1lim f(x), limf(x) 1

18. f : R R x f(f(x)) f(x), x R

+

+ +

1

2x 1

(f f ) (x) ( 1)f(x) lim , N(x 1)

19. x 12

limf(x)= 2

f(x)-g(x) x -1

( )

x 1limg x

20. ) 2x lim f (x)= 0

, :

x lim f(x)= 0

) ( ) ( )( )2 2

xlim f x +g x = 0

( ) ( )x xlimf x ,limg x

.

9

18

( )

21. ( )( )

( ) ( )

x

x

limf x = 0

limg x = 0

f x >0,g x >0

( )x 1limg x

.

22.

( )

x 0 x 0 x 0 x 0

2x 0 x 0

-15 2 3 i) lim ii) lim iii) lim iv) lim -1 2 3v) lim vi) lim

+5x 3+4 -2

23. )

( ++2x 0

x) (x)limx x

) 0

limx 2

2 1 - 1 - x x

)

+ 2x 0

x 25 5lim x

v)

4 4

x 4

x xlimx 1

v)

+ x 0

1 x 1 xlimx

24. : 0

limx

x + 2x + ... + xx

= 28.

25. : 0

limx

x 2x ... xx

= 120

26. )

x 0

1lim x x

)

x 0

1lim x.x

27. 3 3 3 3f (x) 8x x x x 0

f(x)limx

28. 2 3 3xf(x) x x. , x Rx

. x 0

lim f(x)

29.

+ = , , =2 2x 0 x 0

xf(3x) f( x).(ax)f(x)lim 3 R lim 7x 4x x

30. f(x) 2 1x , x Rx

=x 0lim f(x) a , .

9

19

. ..

31. )

+ = 2x 2lim (f(x) x x 5) 7,

x 2lim f(x)

)

= = +2x 1 x 1g(x)lim(x 1)f(x) 5 lim 4,

x 3x 2

x 1lim f(x)g(x)

32. )

= + = ,

x 2 x 2 x 2 x 2lim[f(x) 2g(x)] 3 lim[2f(x) g(x)] 6 lim f(x), lim g(x)

)

( )( )

( ) ( )x 0

3x 0

f xlim =1

6x

lim 1+x -1 g x =10

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

f xlim ,limf x g x

g x

)

( ) ( )

( ) ( )x 0

x 0

f x 3x-xg xlim =5

4xf x 3x+xg x

lim =74x

( ) ( )x 0 x 0limf x ,limg x

33.

= =

+ + x 0 x 0g(x)f(x)lim 1 lim 5

1 x 1 x 3x 4 2

x 0

f(x)limg(x)

34. f(x)+f(x-)=0 , a

= ,

x x 00. lim f(x) 2 lim f(x)

35. f R , f(+) = f( )+f() , R

=x 0

f(x)lim 4x

x

f(x) f()lim , Rx

36.

+ =2

22x 0

f (x)lim g (x) 0x

,

= =x 0 x 0lim f(x) lim g(x) 0

37. ()= + + 3 2x x x, 0

= =x 0 x 0

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

x 0

(f f)(x)limx

38. + 2 2f (x) 2f(x) x 0 R ,

=

x 0lim f(x) 1

.

39. + + =

2x 2ax 5f(x)x 3

, R

=x 3lim f(x) 12

40. ,

+ =x 1

x x 2lim 5x 1

9

20

41.

+ + + =

3

2x 1

kx ( 2)x 4lim L R(x 1)

, ,, L.R.

42. ( )3 2

2

x + 8x + 3x - 2f x =x -

.

f (0,1/2 ) ( )x 2limf x =

,

43. ( )2 2

1x2

2x + 5+2 x+-1lim =151x -

2

, ,R

44. f R x 0

f (xlim )x

=2 .

: R x 0

(lim 42

2

2x)f(x)+x x-3xf(x)

=

. 0

45.

) ( )

+

2x 22x 3limx 2

)

x 1

3x 2limx 1

)

x 0

2 1limx x

46.

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

x0 = - 2 ) f (x) = 3 2x + xx + 1

x0 = - 1

47. i) 2 2x 22 xlim -

x - 4 x -2x

ii) x 0

x -1limx

48.

0

.

9

21

) 2

x 0

cx +2x +dlimx

) 2 2x x - - x -

limx -

) 2

2x 2

x +(b+2)x -16limx - 4

iv) 2 2 2

x 0

c - c - xlim , c 0x

49. , R

+ + + =

2

x 1

x x 2lim x 4x 1

50. , , R ( )

+ + + + =

3

2x 2

x ( 1)x 2lim 6x 2

. ..

51.

+=

x 2

1 xlim

f(x)

x 2lim f(x)

52.

= ++

2

2x 1

f(x) (x)lim(x )

x 1lim f(x)

53. f R ( )

x 3lim x 3 f(x) 2

= m

2 3 2

3 2x 3

(m 1)f (x) f (x) 3lim 2.mf (x) f (x) 2

+ +=

+ +

54. f R

= +x

f(x)limx

+ + 2

x 0

1f(x) bf(bx) f(x )xlim

x,

55. f g R >f(x) 0 >g(x) 0 x R .

+ =

0x xlim (f(x) g(x)) 1

= +0x x

f(x)limg(x)

0x xlim f(x)

0x xlim g(x).

56. = 1 2g (e )

= xg(x) e . pC (0,0)

= x 0

P(x)lim 1x

=x 1

,P(x)lim 1x 1

)

9

22

)

+ + x 0 2

P(x) x 2limx 1 1

x

y

0

y

x 1

1

58. f. 1. f .

2. , ,

.

57. f . ) :

1lim

-x f (x),

1lim

x +f (x),

limx +

f (x), limx

f (x).

)

1

limx (x) f

1 ;

9

23

) ( )x 1limf x ) ( )x 3limf x ) ( )x 5limf x ) ( )x 7limf x ) ( )x 9limf x . 4 . 7 B3. , , .

) ( )x 21lim

f x )

( )x 61lim

f x ) ( )( )x 8limf f x

9 B4. f . .

3 B5. 0x f

( )0f x 0 = . 2 59. + .

) f(x)= - 3x 4 + 5 x 2 +1 ) ( )4 3x +lim 3x - x + 2

) ( )3 2x -lim 2x + 4x + 3x +1

60. + - . ) 3 2f (x) (2 1) x ( 1) x 3x 2 = + + f(x)=( 2 3 22) x 2x 6x 7 +

61. 3 2 3

3 5x + x -

4x + x - 2 3x - 5xlim ) lim3x + 5 2x +1

)

62. + - .

i) 2

2

x x 5f (x)

x x 3

=+

ii) 2

2

x 2x 3x 1f (x)

4x 1 x

+ +=

+

63. ( )( )

3

2x -

-2 x +2x -1lim

-1 x +3

64. ( )3 2

2

x -4x +3f x = -x+x +4

( )lim f x = 2x -

65. R . )

5 4

4 3

(2+4)x +2x -5f(x)=(+3)x -7x +3

.

III.

9

24

) 22x +6x -2f(x)= ( +x -1)3x -2

) 2 3 2

2

( --2)x -(-1)x +3x -9f(x)=(-1)x +3x -7

.

66.

( ) ( )2 2 2x + x -lim 4x + 5x - 3 - x +1 , ) lim x + 4x + 3x)

67.

) 2 55

7 67x +

x +3x -5 + x -4x+2limx -8x +2 +6x -3

) 2

63x -

4x +2x+6 +7-2xlimx -4x+9 +x -7

68. ) 2 2

x +lim( x +1- x -1)

) 2xlim [x( x 1 x)

+ iii) 2xlim ( x 2x 5 x 2)+

+ + +

iv) 2xlim ( 9x 5x 7 3x 2)

+ + v) 2 2

x

x 2x 5 x 4x 2limx 2 x 7+

+ ++ +

69.

) 2 2xlim ( x 4x 5 16x 5x 3 5x 2+

+ + + + + ) 4 2 4 3xlim ( x 6x 1 4x 4 9x 2x 2)

+ + + +

) 2f (x) ( x 4x 1 2 x)= + + iv) 2 2f (x) ( x 3x 1 4x 3 x)= + + + 70. 3 2 24x 5x 3 f (x) 2x 3 4x x 6 + + ,

xlim f (x)+

71. , R 2

xlim ( x 2x 3 2ax ) 3+

+ + + =

72. > 0 4 4 2(3 x )f (x) 3x x 1+ + ,

xlim f (x) 3+

=

73. x

xf (x) 2x 3f : (0, ) R, lim 6x 4

m+

+ + =+

xlim f (x)+

74. x x

f (x)lim 2, lim (2x x)g(x) 5.x x+ +

= =+

xlim [f (x)g(x)]+

IV. O

75. ) x

1 1lim ( )x x

m+

) 6x2lim (x )x

m+

9

25

)x

xlimx

m

v) 3 2

x 3 2

15x 2x 6xlim 64x 5x 12x

m

m +

+ +

76. 2x

4lim ( 16 x 4)x

m

+

77. ( )x + x +

) lim ) lim 2x+

78. 2

2 2x - x +

2 +) lim ) lim -

79. 2

x + x - x +

1 1 1) lim , ) lim x . , ) lim x.

80. :

) 2x.xlim 2x + x +1 )

13 2x -2x +1 +x+2x

lim 2x + x - x+2006

V.

81. )

x x

x 2 x 1x

2 4 3 2lim3 2

++

) x 2 x

x 2 x 3x

4 3lim6 4

+

+

++

, > 0

) x 2 x

x xxlim

+

+

v) 13lim

1 3 ++ +

x

xx v)

1

14 3lim4 3

+

+

x x

x xx

82. f (x) = ln 2x

3 -x , : ) f

) + x

lim f (x), x

lim f (x), 3 x

lim

f (x), 0 x

lim

f (x).

83. f (x) = n

+x

x 22 , > 0. )

f. ) 0 x

lim

f (x), + x

lim f (x). )

f (x) lnx > 0 + x

lim (f (x) - nx).

9

26

84. 12% x

f (x) = 16x 362x 3

++

. ) :

f (x) = 8 + 212x 3+

. )

8%. ) , ; 85. () = + -1-1 + -2-2 + + 1 + 0

Q() = + -1-1 + -2-2 + + 1 + 0

( ) ( ) ( )H x Q x = , i)

(,+ ). : -1 -1x

-lim H(x)=+

.

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

2

( ) ( )lim 2, lim 3, , 0 +

= = + +x x

f x f xx x

.

f(x) ,. (University of New York).

87. ) f: 2 2 2 2f(t)= t +(x + y )t+2 - t +4yt+3 , (,y) .

lim ( ) 0,

=t

f t .

) f: 2 2 2 2f(t)= t +2xt+3x - t -2yt+5y ,

(,y) .

lim ( ) 2,

= t

f t . )

.

88. () limx 2P() = 3

-2+1

limx x02

P() -2+1

=3, 0.

89. f: : 3 3f (x)+f(x)= x x R

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

iii) 0>x ( )

9

27

90. g 2x +ax +g(x)= , R

x -1

1

lim ( ) 2

=x

g x f:

3 2 2 20

( )lim f ( ) 2 ( ) 4 ( ) 8 0l m

= + =x

f x x xf x x f x x xx

x.

) , . ) , , .

91. R)0,(:f xxxxx5x2x)x(f 2247 +m+++= (1) () )x(flim

x () 2x x

)x(flim

92. RR:g,f : 32xx)x(flim 2

2

x=

+

+

2x

)x(glim 2x =+ . R 6x)x(g)x(f

x3)x(g)x(f)1(lim

2

x=

m+++++

+

.

9

28

9

29

- 9

28

.

I. 0

1. 0=0 :

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

1 , x 0

x xx

= ii)f(x)=

1-2 , x 0

2 , x 0

+

=

iii) f(x)= 1 1

, x 0

-2 , x 0

+

= iv) f(x)=

9 5 , x 0

4 , x 0

=

2. ( )

2 1x ,x 0f x = x

1 ,x = 0 : 0=0

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

-1 , x

+

= =

< 0 , f(a) 0 ,

0

0

0

00 xa

)x(fx

)xa(f:)a,0(x

=

( )

39. f [0,1] 0 f(x) 1 [ ]0,1 , , [0,1), :f 2005() + = f() 40. f, g [ ], = ( ) ( ) [ ]f g , = = [ ]0x , , ( )( )0 0g f x x= . 41. f = [, ]. : f (+-)

. , , : f (+-) = f(). 42. f = [, ] >0. ,

, : f() = f ( ) f ( )

1 + +

.

42. f [0,4] f(0)= f(4), , [0,4]

- = 2 , f()= f(). 43. f [ ], 1 + : ( ) ( )f f 1 0, . + + = ( )f x 0= [ ], 1 + 44. f: f(1) + f(2) + f(3) = 0.

f() = 0 .

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

.

. BOLZANO

- 9

34

46. f(x) = x2 + x + g(x) = - x2 + x + 0.

1 f 2 g 1

- 9

35

i) 2000 20041 2 0

1 1 + +

+ = +

(-1,1).

ii) 06 4

+ =

( , )6 4 .

56. f(x) = ex- -1 g(x) = ln(x+1) + , 3(0, ).4

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

ii) f() = [,+1],

: xlim g(x) .

=

57. f,g R f(1)+f(2)++f(2014)= g(1)+g(2)++g(2014)., { }1,2,3,..., 2014 f() =g()

58. f ,g [0 ,] . f fggf = [0,] f( )=g( )=

59. 2 2 2

0, ,, 0x x 1 x 1

+ + = +

) (-1,1).

) 1 2, 2 2

21 2

1 1

+ = .

60. f ( ) 3 2f x x x x = + + + : 0 >

0 + + + = 3 2 0 + + > ( )0x 0,1 ( )0f x 0= 61. f : : ( ) ( )f xf x e 5 4x+ = x ( )f 1 0= : ) f . ) ( )( ) ( )3f f x f 5 10x 0 = ( )0,1 62. f(x)=ax3+bx2+cx+d , a>0 , d

- 9

36

63. ax5+bx4+cx3+dx2+ex+f=0 f>0, a+b+c+d+e+f=0,

5a+4b+3c+2d+ e >0 (0,1)

64. f : f3(x)+f2(x)+f(x)=x3-2x2+6x-1,

x R , , R 20 , n N * : xn= .

( ;)

68. f, :

63 x2x+8 - x+4 xf(x) +x6

, x 4

f(0) (0, 1] ,

f() = 6

+ 6 .

69. f , :

2

x 1

f (x)lim 4 4(x-2) (x-2)f(x) x 4 x .x 1

=

Cf

= x2 - x + 1

(1,2).

70. . f 11 . , ,

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

- 9

37

. f 11 ,

.

( ).

.

f R f(f(x))=-x

. A f R f(a)+f(b)=f(c)+f(d) a,b,c,d

f

71. f R f(x).f(f(x))=1 , f(9)=1/9 f(1) 72. f R ( )f 8 7= x

: ( ) ( )( )f x f f x 1 = ( )f 4 .

73. [ ]f : 2,10 , ( )f 2 10= ( ) ( )( )f x f f x 5 = ( )f 10 ( )f 5 . 74. [ ]f : 1,2 . ) ( ) ( ) [ ]x 1g x f x e , x 1,2= + . ) [ ]0x 1,2 ( ) ( ) [ ]0x0f x f x e x 1,2< + . 75. [ ]f : 2,10

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

0

3f 3 5f 6 2f 8f x

10+ +

=

76. f [ ,] , 1 , 2,, [,] 1 , 2,, R+

(,)

kkk

xfkxfkxfkf+++

+++=

...)(...)()(

)(21

2211

77. f [0,1] f(0) = 2 f(1) = 4. f , (0,1)

f(): =

1 2 3 4f f f f5 5 5 5

4

+ + +

II. .. -...

- 9

38

78. f: [ ], . ,

32-4000+2001 = 0, [,]

: ( )2 2f f f 2000f3 2 2 3

+ + + + + + =

.

.

78. f f 2(x) + 2= 5

= (0,5). f 1.

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

79. f : f 2(x) = 2exf(x),

x

80. f : :

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

81. f : :

f 2(x) 2(f (x)x = 1, x . 82. H f R ,f(0)=-2 2f ( x ) xf ( x ) 4 0 , x R+ =

f .

83. f : ( ) ( )2f x 1 2f x x= + x .

84. f,g: [0, + ) g(0) = 1

x[f(x)-g(x)] = x [ 3f(x)+g(x)]. ) f(0). ) [0,4] f(x)0, :

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

2. g(x) > 0 [0,4].

- 9

39

85. f [1, 4] , f(x) 0, x[1, 4], f(1) > 0 f(1)f(2) = f(3)f(4) . :

. f(x) > 0 , x[1, 4] .

. g(x) = f 2 (x) f(1)f(2) [1, 2] .

. f .

86. f,g : R R f(x)g(x)=ex , x R

f(1)>3 g(2)>2. N : ) f(x)>0, x R

) x0 (1,2) g(x0)=x0

87. f R , f(0)=5 2)x(f Rx

f .

88. H f R , f(0)=-1 f

.

89. [ ]f : 3,3 ( )3 4x f x 27+ = [ ]x 3,3

f ( )3,3 . 90. f, 4x 2 + 9f 2 (x) = 36 , x(3, 3) .

f(0) = 2 .

91. ( ]f : 0,1 ( ) 1f x lnxx

=

) f. ) f. ) ( ]0x 0,1 : 0 0 02x lnx 2 3x= 92. f(x)= 4- x - 2+x . . f . . f. . f(x) < 0 .

IV.

- 9

40

93. f (0, +)

+x 0lim f(x)= R

x +lim f(x)=R,

x0 > 0 , 0x +10 0f(x )+e +ln(x )=1 .

1. f : ( ) += +2x 1f x 3e 5x 3 . ) f. ) f. ) ( ) =f x 0 R.

2. f : ( ) 20112 5 7,f x x x x R= + . i. f R . ii. ( ) =f x 0 . iii. f .

3. f ( ) = +xf x 4 e 2 3 . i. . ii. . iii. 1f .

4. f ( ) ( )= + +f x 2ln x 1 1 3 . i. f. ii. f 1-1. iii. 1f . iv. ( ) + =1f 1 x 2.

5. f ( ) = +

x1f x 3x 22

.

i. f. ii. x R 2011. iii. : +

- 9

41

7. ( )x 1

limf x , :

i. ( )

= +x 1

2x 1limf x

ii. ( )

= +x 1

f xlim

4x 3 iii. ( )( )

+ = + x 1lim f x 3x 4

8. f : 1,5

( )A 1,8 ( )B 5,12 . i. f .

ii. f 293

.

iii. ( )0x 1,5 : ( )( ) ( ) ( )+ +

=02f 2 3f 3 4f 4

f x9

9. f ( ) ( )= + xf x ln 3e 1 2.

i. f. ii. f . iii. 1f . iv. ( ) ( )< 1f x f ln5 2 2.

10. f ( ) = 3f x 2x 3x 1. i. f. ii. f . iii. ( ) =1f x 2 iv. ( ) 1f x x 1

11. : Rf R : ( )( ) ( )+ = +f f x f x 3x 2 x R .

i. ( )1f 1 . ii. ( )f 3 . iii. ( ) =1f x 3

iv. ( )( ) ( )

+ ++ x

3 x x xlimf f x f x 2

.

12. R f :

( ) ( )

+ =

2x 1f x x x 1

lim 2x 1

i. f ( )M 1,1 .

ii. ( )

2x 13f x 2 1

limx 1

.

13. f ( ) += +

x 1f x 2ln 31 x

.

i. f. ii. f .

- 9

42

iii. f 1f . iv. : ( )

x 1limf x ( )

x 1lim f x .

14. : Rf R g ( ) +=

x 2g x ln2 x

.

i. f g . ii. h : ( )( ) =h g x x . iii. h .

15. R f g : ( ) f x 0 x R . ( )A 2, 1 . = 1 1 =2 5 ( ) =g x 0 . : ) f R . ) ( ) 0

: + =

4 1 3 1ln2 2

17. : Rf R : ( ) ( ) = 32f x 3 2x 3f x , x R . i. R. ii. f R , f 1f . iii. ( ) =f x 0 . iv. f 1f .

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

18. f 3,3 ( )+ =2 23x 4f x 27

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

- 9

43

iii. f.

iv. ( ) =f 1 6 ( )

x 0

3 3f x2lim

x.

19. [ ): 0,f R+ :

( )+ + + + +2 8 2 xx 2x 9 3 xf x x 3x 3

>x 0 . :

i. :

+ + 2

x 0

x 2x 9 3lim2x

. ii. :

7

x 0

2lim xx

. iii. : ( )x 0

limf x .

iv. ( )f 0 .

20. : Rf R : ( )( ) ( )+ = +f f x 2f x 2x 1 x R ( ) =f 2 5 . i. ( )f 5 . ii. f . iii. ( )1f 2 . iv. : ( )( ) + =1 2f f 2x 7x 1 2 .

21. : Rf R ( ) = + +2 2x xf x 2 1 ,x R R . i. f R. ii. ( ) = f 0 2 f.

iii. : ( )+

+

x

x xx

2f x 3lim , 3

3 2 4 3.

22. : Rf R : ( )+ +4 4x 1 4f x x 2

x . i. : ( ) 1 1f 04 2

( ) 1 3f 1

2 4 .

ii. :

4

x 0

1lim x fx

. iii. :

+

+

5

2x 0

1x f 4 3xxlim

2x 3 x.

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

23. i. ( )

=

x 0

2f x 4lim 2

x ( )

x 0limf x .

ii. : Rg R : ( ) + +xg x 2 2 x x x x . ( )

x 0limg x ,

. iii. : ( ) ( )( )

+

+

2 2 2

2 2x 0

x f x 2xlim

x x g x

24. : Rf R : ( ) ( )+ = +33f x 2f x 4x 1

x R .

- 9

44

i. f 1f . ii. 1f . iii. f 1f , =y x . iv. : ( ) ( ) = x 1f 2e f 3 x .

25. ( ) = + f x x 1 1 ( ) = g x 2 x . i. f g. ii. f g . iii. f 1f . iv. f f g .

26. f ( )

+

2

2

2x x , x 0x x

f x ,x 0

8x x 16 3x, x 0

i. , . ii. : ( )

+xlim f x .

iii. : ( )xlim f x .

iv. ( ) ( )= +f x 2ln 8x 1 ( )0,1 .

27. f ( ) ( )( ) ( )

( ) ( )

+

=

+ +

2

3 2

2

x 5x 6 , x ,0 0,24 x 2x

f xx 1 , x 2,

2 x 4

{ }: R 0,1g R

: ( )

+=

x 0

x g x 2xlim 5

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

: i. ( )

x 2limf x . ii. ( )

x 0limf x . iii. ( )

x 0limg x .

iv. ( )x 3

limg x .

28. f ( ) = + + 3xf x 3ln2x e 4x 2 . i. f. ii. : ( )

x 0limf x ( )

+xlim f x .

iii. ( ) =32f x e .

iv. : ( ) ( ) ( ) + + + =

23 12 2 63ln4 3ln 2 2 4 1 e e 8

- 9

45

29. : Rf R : ( ) 213 x 2xf x x

2, x R .

( ) ( )+ + = 24f x 3f x 1 2x 2013 , x R . i. ( )

x 0limf x .

ii. ( )f 1 . iii. f ( ) = g x x 1 ( )0x 0,1 .

30. : Rf R : ( ) = + + 2 2 2x x f x 1 x

x R

1A 0,2

.

i. . ii. =1 =1 f.

iii. : ( ) x 0

f xlim

x.

31. f ( ) + =3 x x

xx 2 3 2 4f x

2.

i. f . ii. ( )

xlim f x .

iii. ( )+xlim f x .

iv. ( ) = f x R R .

9

46

. 0f x( )

1. : i) 43x)x(f ++= x0 3= ii)

3 2x)x(f = 0x 0 =

iv) x3xx)x(f += x0 0= v) 11x)1x()x(f += x0 1=

vi) 3 4x)x(f = 0x 0 = vii) 3 x)x(f = x0 0=

2. x0

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

f x ,x 02 x

= =+

3. ( ) 0xf x ,x 0

1 x= =

+

4. ( ) 0f x x x , x 0= = 5. ( )3 2

0f x x , x 0= = 6. ( )2

0x , x 0f x , x 0x

0 , x 0

= =

=

3. 0

) 12x , x 0f (x) x

0, x 0

==

, 0=0 )

13 xx e , x 0f (x) 0, x 0

1 12 xx e , x 0x

, 0=0

4. f 0 2 4f (x) x x x , x R ,

f 0.

5. f , : 22 xx)x(fxx + x R , f 0x 0=

f 0( )

6. f , g , f()=g() f(x)+x2 2g(x) , x R, + g ( ) f ( ) 2 =

7. f x0 ,

g 00 0 0 0

f (x) , x xg(x)

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

=

+ > x0 .

9

47

8. g,f : 2g(x) f (x) g(x) (x 1) + Rx g '(1) 2= . f 1x 0 = )1('f .

9. , x = 0 ,

23x 1,x 0f (x)

ax ,x 0

+

9

48

18. 0 f ,

3 2 2 3

xf x 2f f x 3f f x 2f

x +

( ) ( ) ( ) ( ) ( ) ( )lim

19. > 1 2

0 0 1 2h 0

f x h f x h h h mh

+ + + + + =

( ) ( ) ...lim ,

.m R 0 1f (x ) m=

20. f(3) = 10 f (3) = 6 , 3x

lim2

2

(f(x)) -100x - 9

.

21. f 2)1('f)1(f == . :

)2

2x 1

f x 4x x 2

+ ( )lim )

x 1

xf x 2x 3 2

+ ( )lim

22. f Rx 0 , :

0

0 00 0 0x x

0

x f x x f x f x x f xx x

=

( ) ( )lim ( ) ( ).

23. f R R: *+ 1)(f)(f == :

x

f x f xx

( ) ( )lim

24. f R, x0 0= f(0)= 0 ,

: )0(fx1fxlim

x=

+.

25. f (). ( ) ( )x

xf f xlim

x

26. f (). ( ) ( )x

x f f xlim

x

2 2

27. f (). ( ) ( ) ( )x 0

f x f xf lim

2x+

='

28. : f 0x , :

) 0 0 0h 0f x 2h f x 2f x

h+ =( ) ( )lim '( ) ) 0 0 0h 0

f x h f x 3h 4f xh

+ = ( ) ( )lim '( )

)2 2

0 00 0h 0

f x h f x 2f x f xh

= ( ) ( )lim ( ) '( ) v)2

0 0h 0

f x h f x 0h

+ =( ) ( )lim

III. 0f x( )

9

49

29. f R i) )(f2)(f

x)(fx)x(flim 2

22

x=

ii) g R , :

)(g)(f)(f)(gx

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

=

IV.

30. f x y R, :

22 y)x(f)yx(fy)x(f ++ . f x R0 .

31. f f(x + y) = f(x)f(y) , x, y f(0) 0 . f 0 ,

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

32. f, g R : i) f x y f x f y+ = ( ) ( ) ( ) x y R, . ii) f x 1 x g x= + ( ) ( ) x R iii)

x 0g x 1

=lim ( )

f x R0 .

33. f : 2f 2 h 1 2h h 3h+ = + + +( ) , h R ,

: i) f 2 1=( ) ii) f 2 3 =( )

34. f 1 f( x ) = f( x)+ f ( ),

f xf x f 1 x= + ( )( ) ( )

35. A f xy f y f x x y 0 f 1 1 2= + =( ) ( ) ( ), , ( , ) , ( ) / f

(0, )+

36. f : R R , : yxf (x y) e f (y) e f (x)+ = + , x,y

f (0)=2

) f (0) 0= 2x ,f (3x) 3e f (x)= x R

) x 0 x 0

f x f 2xx 2x ( ) ( )lim ,lim ) f(x)

9

50

37. N

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

3. x

f(x)=x

4. f(x)= x

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

6. 43f(x)= x

38. f(x) = 2

2.

x x x, x

9

51

4xy (x)+2y (x)- y(x)=0 ,x R

46. f(t)=e-t(t), , R* , :

f (t)+2 f (t)+(2+2)f(t). (

.)

47. : y axay (x)=e y (x)-3y (x)+2y(x)=0 , x R .

1 1 2 2g(x)=c y (x)+c y (x)

48. f R ( ) ( ) + =2 2 4f ( x ) f ( x ) x

f (0)= 0

49.

) xf(x)= x , x >0)

2x +3xf(x)=x

) xexf(x)= x ,x >0) ( ) ( )xf (x) x ,x 0,=

50. f : R R . g ,g ) 2g(x)= f(3x)+f (2x)

) ( )3 4 2g(x)= f x +x +1 ) 41+f (x) f(x)g(x)=e -x ,x >0

51.

) f(x)=(2x)(2x) ) f(x)=4(ln56xe ) ) f(x)=xx , x>0

) f(x)=1x(1+x) ) f(x)=

4 2

21 ln x+

) f(x)=3 x 2 , x [2,11)

3 x 2 , x [11, )

+ +

52. R f ( )6 2 3f(x )= f x f (x) 0, f(1)

53. / R : x

1 1 1- =f(x) g(x) e

: 2 2g(x)-g(x) f(x)- f(x)=

[g(x)] [f(x)]

54. = + = >5 3x t 5t , y 5t ,t 0 dxdy

55. ) f(0) = f (0) = 0 ) = h(x) f(x) f (x) 0, x R

9

52

= +

h (x) f (x) f (x), x Rh(x) f(x) f (x)

56. * 1x + 2(2x) + 3(3x) +.+(x) x x 1,2,3,. ( ),

: 1 + 22 + 33+.+ 1 .

57. A f x : f(ex-1)=(x) , f (1) , f (1)

58. f(x)=x(x-1)(x-2) (x-100) , f (0)

59. A f(x)=(x-)3(x-)5 , f (x) 3 5= + x x f(x) x- x-

60. f , g , g(x) 0,

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

. A 0h (x ) 0 = :

h(x0)= 00

f (x )g (x )

, g(x0) 0

61. f . : ) f , f ) f , f

62. f ,

g(x) = f(f(x))-f 21

x 1

.

63. f,g x0 f(x0) 0

G(x) = f(x)g(x)e

G(x0) = 0 g(x0) = f(x0) g(x0).

64. :

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

)

65. ) xf(x)= , x R2

) *1f(x)= , x Rx

.

66. (x) = [(x)]2 x .

9

53

o (x)- (x) = 2x2-7x+7, x .

67. P(x) : ) ( )xe P(x) =e x x(x-1) , x ) 1 [ P (x)]2=P(x) , P(1)=0

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

68. f(x) = x2+x+, 0, 1,2, : ) f( 1)+ f( 2) = 0 ii) f( 1)f( 2) 0 ) 1/f( 1)+ 2/f( 2) = 1/.

69. : ) = 1 + 2x +3x2+.+x-1, x * ) = 2x + 4x3 + 6x5 + + 2x2-1, x *. 70. f(x) 2, : ) f(x) = (x-)2(x)

f() = f() = 0 ii) ,

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

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

i) i1 2

f (x) 1 1 1= + + ....... + , x i=1,2,3, .... ,f(x) x- x- x-

ii) f(x)f (x) < [f (x)]2 , x

iii) N 1 2

1 1 1+ + ....... +

iv) f(1)= f (1)=0 f (1)0 1 2 1 3 1

1 1 1+ + ....... + 0 - -

=

72. : ) f: (0,) R , f(x) = x, ) f -1 (-1,1

[f -1(x)]= 2

1 , x1 x

(-1,1).

73. f: ( ) ( )0, 0,+ + , f(x)f 1x

= x,

>0 :) f 1x

= 1/xf(x), x 0.

) [f(x)f 1x

]=0, x0.

IV.

9

54

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

1f

75. :f R R 1-1 R ( ) =f R R

R 1f (f ( )) 0 = 1f

0x =

76. f,g 1-1 R.

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

( ) ( ) + = + f g g f (x) x f g (x)

77. f(x)=5ex+x-2 . N

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

) ( )1 1f (3) 6

=

78. f ( )+0, , ( )0, + = + .f( ) f( ) f( ) ) =f(1) 0

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

79. f R , + = +xf(x ) e f( ) e f(x) + .+ , R

) f(0)= - = 0 ) xf (x)= f(x)+ f (0)e + x

80. f(x)=exln(2-ex) x0=0

81. Cf x0:

) f(x) = (x2-1) x , x0 = 4 ii) f(x) = x

x-xx, x0 = ) f(x) =

ln xx

, x0 = 1

iv) f(x) = xxe

, x0 = 0 v) f(x) = xx , x0 = 12

V.

9

55

82. f(x) = x3-4x, Cf xx.

83. Cf f(x)=2x 5

x 1

,

M (x0,f(x0)) f(x0) + f(x0) = 4

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

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

85. f(x)=x3+x2+x (0,1)

86. Cf f(x) = lnx

87. Cf f(x)=x3 o (0,-16).

88. Cf f(x)= xx 2+

,

(0,1).

89. f: R R : 3f(x+1) 2f(x-2) = x2 + 14x 5,

xR. ) f. )

Cf , (1,-14

), .

90. f : f (lnx) = x . lnx x, x > 0.

. Cf .

. Cf x0 = 0

91. (,) 2

c: = x2. ii) : = - 14

,

.

. 0

9

56

92. f(x)= 21 x ,

x x (,f()) , 0 Cf

) Cf (,f())

) :

i) =3 Cf

ii) =1 Cf

93. f (x) = x2 x + 1 ( ),

: ) = 3. ) 45 xx. ) y = x +3 .

) y = - 21 x + 3.

) xx. ) yy.

94. f (x)=x+ln(x-3)2 i) =3

ii) 45

iii) yy

95. y=3x-8 f(x)= x+ln(x-3)2

96. f 2f(x)=1+xf3(x) , x y=-x+2 Cf

97. f(x)=x2+x+2 : y=-x+ Cf

98. 0 1< , : y=x Cf f(x)=x.

.

. Cf

IV.

9

57

99. g(x) = f(x)(x), 0, f R f(x) 0 xR. (x0,0) Cf Cg,

Cf Cg (x0,0).

100. g(x) = 1x 1

f(x) = x2 x +9. ,R,

Cf Cg 2.

101. , , f(x)=+xx g(x)= x2+x+2

(1,2)

102. f,g, : f R f(x) 0 , xR

i) R ii) g(x) = f(x)(x), xR

) [ (x)] 2 + [(x)] 2 = 1, xR (x0,0) Cf Cg, Cf Cg

103. N f(x)= 1x

g(x)=x2-x+1 ,

104. A f , f(x) 0 x g(x)=f(x)3x , f g

V.

105. g(x)=2x2x+ f(x)=23x 2x 4

x + .

, Cf Cg

2.

106. 4(1-x)e2x = 1 (0,1). ii) N Cf Cg , f(x) = ex g(x) = x .

107. = xf(x) e = g(x) lnx fc

9

80

gc x x, f g

108. f(x)=ex g(x)=-x2-x. N Cf (0,1) Cg

109. Cf , Cg f(x)=x g(x)=x3-3x2+4x-1 , Cf , Cg

110. . f(x)=4-x2 g(x)=-x2 +8x-20.

V. 111. f : f (4) = 0 (f (x))2 = f (x) x R, ) f. )

Cf y = - x + 1

112. f [0, )+ x R = + +x 2 .f(e ) x 2x 3 fc A(1,f(1))

113. : = 2+1

f -1 , ( ).

x -x +1lim f +1 x -22- x

114. : =2-4 ( )g(x)= ln f(x) 0x = 2 f

2x f(x)-4 , x 2h(x)= (x -2)f (x)

6 , x = 2

115. f , g , h: . f(x) 0 , g(x)=f(x)h(x) x , f , g , h h2(x)=1-[h(x)]2,

Cf Cg .

116. f: x f(x)=f(+x) , . Cf

(-,f(-)) (+,f(+))

9

81

x=.

117. f f (1)=1 g g(x)=f(x2+x+1)-1 , x

Cf (1,f(1)) Cg (0,g(0))

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

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

119. . cm r = 4-t2 , t o sec N

V t=1sec

120. x2 + y2 = 4. (-1, 3 ), 6 m/sec.

.

;

121. x >0 , (0,0), (4x,0) (0, x -2). x

2cm/sec, x = 9 cm.

122. 2m/sec 12m.

123. x2 + 2 = 2 xt

= .

t

0.

; 124. f(x) = x2 2x.

9

82

T t0 (2,0) x 3 cm/sec.i) (t) t0. (t) x (t) 0 = , = f(x).

125. (t) = t40

1 19.2+,

t . (t) = N (t) = 10-3, .

126. 170cm

300cm. 7,2Km/h

127. 9 cm2. O

10 cm2/sec.

16 cm2.

128. 13m . 2m/sec. :

.

5m.

.T 12m.

129. f(x) =lnx, x0 (,ln), 0. Cf .

;

v = 2m/sec,

t t0

.

130. f(x) = x3, (,3), >0, yy v = 3 m/sec.

i) Cf . ii) Cf

.

9

83

iii) ( ) .

iv) t t0 yy

2m.

De L Hospital

131. ) 2x 0

x x ln(1 x)lim

+ +

) x 4

2limln( 2 )

)

x x

x 0

e e 2xlim1

v) x x

x 0

e elimx

132. :

) 4

2x 0xlim

x +2x-2 )

2

xx 1ln xlim

e ex ) 2x 0 x

x-xlimxe -1-x- 2

) 2x 0

xx-x+1limx

) 2

x 1x 1limln x

133. )

3

xxlim

e+ ) 1x 0

x

ln xlime

) xx

xlim , 0, 0e

+ > > v)

2

3 2x

x ln xlimx x x 1+ + + +

134. :

) x

ln( x 1)limln( x 2)+

++

) x

x

elimx

+ )

2

xx

x x xlime x 2++ + + +

135. )

x

1lim ln 1x+

+ ) ( )

1lim ln ln 1

x x x ) ( )x 0lim ln x+

136. :

. 00

.

. 0 ( )

9

84

) 32x 0lim (x ln x)

+ ) 2

1x

2x 01lim ex

) 12x 0lim ( xln x)

+

137. i)

x 0

1lim

ii) ( )xxlim e ln x+ iii) 2 2x 0

1 1lim

iv) x 0

1 1lim2 1

138. :

) x 1

1 1limln x x 1

) 2 xxlim (x ln x e )+

+ ) 2 x

x 0 x elim -

x x

139.

) ( )x 0lim

+

)

x 0lim

+

)

11

x 1lim

v)

2

x

3lim3

+

+

v) ( )3

x 0lim 3 2

+

140. 2xf (x) ,0

g(x)0, 0

< < =

f R

f(0)= .f (0) f (0) 0 = = Na f g

141. f R f(0)=0.N xx 0

f (x) f ( x)lim 2f (0)e 1 =

142. f R f

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

2

xx 0

f (x)lim 1(e 1)f (x) + =

f (0) 2 =

143. x ln x x 1,0 x 1f (x) 1 x2, x 1

+ < =

= f

. ( ) ( )

V. ( ),1 , 000

9

85

1f (1)2

=

144. 2x 2(1 e ) ln x , x 0f (x)

0, x 0

==

)

2x

2x 01 elim

x

2 2x 0lim(x ln x )

) f

=0. ) f (0,0)

145. f 0 =0 (0) 0.=f f (x)

x 0lim x 1

+=

146. f : R R f(x)>0

x

.lim (f (x) f (x)) l R+

+ = lim ( )x

f x l+

= lim ( ) 0+

=x

f x

147. f ( ) 0, f

2x1 1 f ( )lim

(x )f ( ) f (x) f (a) 2f ( )

=

148. x x xlim f (x) lim f (x) lim f (x)+ + +

= = = + x

f (x)lim 1xf (x)+ =

2

x

x f (x)lim 0xf (x)++ =

149. f: xf(x)+ex=f(x)x+ex x . f(0).

150. f: (1-x)f(x) = ln(1+x)-x x -1. f(0).

151. i) N 1+lnx = x (0,2). ii) Cf f(x) = xlnx+x+2, x(0,2) (2,5) f((0,2)).

152. f : . : 2h 0

f (x 2h) 3f (x) 2f (x h)lim 3f (x)h

+ + = .

ROLLE

9

86

. ROLLE

I.

153. Rolle ) f(x)=2x3+x2-8x+1 , x[-2,2]

) f(x)=2x+2x-1 , x[0,] ) f(x)=x2-4x+|x-2|+3 , x[1,3 ]

154. f(x) = 29x 14x 9, x 1

x+1 , x 1

+

.

.ROLLE [0,2] f.

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

. , R

f . ROLLE [-,1] x0(-,1) f( x0) = 0.

156. 2

3

x a x 0f (x)

x (-1), 0

+ =

+ > , R

f Rolle [-1,1] 157. : i) f(x) = xx

. ROLLE [- ,2 2 ] ii) xx = 1

(- ,2 2 ).

158. f : [,] (,) f() = f() = 0 :

a) g(x) = f (x)x c

c[,] . ROLLE

[,]. b) c[,], c0(,) , Cf (c0, f(c0)) (c,0).

159. f: [a,]R, [,] (,). : I )

)x)(ax(e)x(g )x(f = Rolle [,]

) (,) , f ()= 1 1a -

+

9

87

160. f,g : [,] f() = f() = 0 f(x) 0 x(,)

: a) h(x) = f(x)e-g(x), x[,] . ROLLE [,]. b) x0(,) , Cg (x0,g(x0)) : f(x0).x - f(x0). + = 0.

161. f(x) = x (x-1) ,*.

x0(0,1) , f(x0) = 0 (x0,0) A B =

(0,0) (1,0).

162. f [,]

f() = f() f() = f() = 0, x1, x2(,) , f (x1) = f (x2).

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

x0(,) , : 02 20

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

=

.

164. f ,g [,], (,) f() = f()eg()-g(), x0(,) , : f(x0) + f(x0).g(x0) = 0.

165. f R 10f(x)6f(5)+4f(3) x, f() = 0. 166. f [2,3] f(3) f(2) = ln3-ln2,

(2,3), f() = 1

.

167. .

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

0

f (x ) f ( ) .x

169. f [0,2] (0,2),

(0,2), f() = f(2-). 170. f [,], (,)

f() - f() = 2-2. x0(,) , : f(x0) = 2 x0. 171. f: [1,4] R f(1) = 2

9

88

f(4) = 8. Cf

172. f: [,] R [,] , (,)

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

173. ( ) ) x1 , x2

f

f

) 1 , 2

f

f

) f x1 , x2 , , x

f -1

: ( ) f :

F

F (x) f (x) = , x

. ROLLE

f F f F

0 c f(x) f(x)

1 x f(x) f(x) 12

f2(x)

x +1x

+1 f(x) f(x)

1+1

f+1(x)

1x

2 x f (x)f (x)

2 f (x)

9

89

x -x f(x) f(x) -f(x)

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

ex ex ef(x) f(x) ef(x)

1x

ln|x| f (x)f (x)

ln|f(x)|

x x

ln f(x) f(x)

f(x)ln

f F f F

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

2

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

f (x)

g(x)

2

f (x) xf (x)x

f (x)x

[f(x)]2+f(x) f(x) f(x) f(x) [f(x)+ f(x)g(x)]eg(x) f(x)eg(x)

174. 0 , , R 03 2 + + = . 2++=0

( 0 , 1) .

175. + ( -1) =0 ( 0 ,1 )

176. 1 02 1..... 0, 1 3 2 1

+ + + + + = +

*,

x+ -1x-1 +..+2x2 + 1x + 0 = 0, (0,1).

177. 8x3+9x2-6x-2=0 (0,1)

178. (x2-4)x+2xx=0 (-2,2)

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

f() f()=

) xf(x)=f(x)

(,) )

Cf

180. ex =1 , ex = -1

9

90

181. 24x+xe = 2x 2 + 4 =

182. 3 3 + =0 (-1 , 1 ) 183. 2 + + =0 184. f : R R R

f (x) 0 R .

f (x) =0 .

185. 0, x4+x3+2x2 +x+ = 0 .

186. 4x3 + 18x + = 21x2 (1,2).

187. f(x) = 3 22 7x - x +3x+3 2

, R.

. 188. x3+x2 +x+ = ex

. 189. x4+2x3+3x2 x + = 0

, R.

190. x2-2x+2ln(x+1)=0

191. x4-x3+5x2+x+=0 ,

9

91

) : (x+3)1996 = (x+1)1996 +16499.

193. 4 + 3 +32 + + =0

, 2 > 8 194. (x) = (2x-1)(2x+1)(x2-3x).

(x) = 0,

..

195. f(x) =

3

2x+2, x -1x - x, x >-1

. f

... [-3,2] , (-3,2) ... 196. f f(x) = x(1-lnx). f

... [1,e] , (1,e) ... 197. f [,], (,)

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

: f( )( )f ( )f ( ) f ( )e

=

. ..

198. f [1,3] f(1) = 2005

1 1f (x)2 2

x(1,3), 2004 f(3)2006.

199. f(x) R f(0) = 0, : f(1) f(1) f(0).

200. f [1,5] f(1) = -2 f (x) 2 x(1,5), : -10 f(5) 6.

. ..

9

92

201. ) f [,] : f()- f()f ()<