28 3391fylladio math_kat_g_lyk(2016-17)
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14
2016-17
-
.
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14
1
1. :
) ( ) 1 1f x x x ) ( ) 5 5g x x x
) 1
( )3
xh x
x
)
2( )
2 15
xc x
x x
) ( ) 3 1d x x ) ( ) ln(1 ln )k x x
) ( ) 2 1l x x x ) 3( ) ln(2 )m x x
2. 2
2
1 , 5( )
ln( 1),
x xf x
x x x
. ,
f .
3. 2
1( )
3 4 4
xef x
x x
. , f
.
4. :
, 0 3
( ) 1, 1 0
2, 3 1
x x
f x x
x x
.
5. :
) ( )f x x ) ( ) | |g x x ) ( ) | 1| 2h x x
,
6.
) 2 5 6
( )3
x xf x
x
) x x - xf
)13 3 3 2( ) x x xf x ) ( ) 1 2f x x
7. f , g 2 22( ) ( 3( ) ) g x f x f x x x ,
g Oy.
8. :f : 3 2 2( ) 2 ( ) 5 ( ) x xf x f x f x e e ,
x . fC xx.
9. :f 2
2
2 ( ) 3 ( ) 1( )
3 4
f x f xg x
x x
xR .
Cg ' .
,
10.
2 2 4f x x x ,
2 2g x x x .
-
14
2
11. ( ) ( 2) ( 2)f x n x n x
x y=f(x)
ln5y .
12. , :f g , : ( ) 1 ( ) ,xf x g x e x
f,g.
13. 3 2 24 4 1, 2 5 3.( ) ( )f x x x x g x x x
) Cf Cg ;
) Cf Cg
= 6..
14. f = g. f g R
f (x) = g(x).
) 3 2
( )x x
x
e xef x
xe
2
( ) 2xe
g xx
) 2
2
4( )
2 | |
xf x
x x
2( ) 1
| |g x
x
) 9
( )3
xf x
x
( ) 3g x x
) 3
1( )
1
xf x
x
3 2 3( ) 1g x x x
15. .
) 2( ) ln( 2) ln(2 ) ( ) ln(4 )f x x x g x x
) 1
( ) ln( 1) ln( 1) ( ) ln1
xf x x x g x
x
) 2( ) 2ln( 3) ln( 3)f x x g x x
16. , , 2
( )2
x xf x
x
2 ( 1) 2 3
( )1
x xg x
x
.
17. 2 9
( )2
xf x
x
2
1( )
6
xg x
x x
.
: f g f g ,f
g.
18. f (x) 1x g(x) 6 x .
f+ g ,f- g ,f g , ,f g
g f.
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14
3
19. 1, [0,2]
( ), (2,4]
x xf x
x x
2, [ 1,1]( )
2 4, (1,3]
x xg x
x x
.
: i) f g ii) f g
20. 1, [ 2,0]
( )2, (0,2]
xf x
x
0, [ 3, 1]( )
3, ( 1,1]
xg x
x
.
: ) f g ) f g
21. f, g A :
( ) ( ) ( ) 4 ( ) ( ) ( ) 4 8f x f x g x g x f x g x x A . : f g .
22. f g x :
2 2 2 24 3 (x) 2 (x)g(x) g (x) 4x (x) ( )x f f f f x f g
23. fog , gof : ) x x xf , g x -x -
) 2
( )2
xf x
x
, ( )g x x
) ( )f x x , g x - x
) ( )2, 1
, 1f x
x x
x x
, g x x 2
) ( ) 2f x x , g x x 1
24. 3
( )x
f xx
fof.
25. 2 3, 1 1
3 1, 1 10f x
x x
x x
2, 0 4
3, 4 7
x xg x
x x
.
fog.
26. f : ) (f og)(x) = x2-4x , x ( ) 4g x x
) (f og)(x) = 51 x , 1x 5( )g x x
) (fog)(x) =1
1
x
x
, 1x g(x) = ( 1)n x
27. g :
) 2f g x x x 1f x x
) xf g x e x x 1f x x
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14
4
) 2 1f g x x lnx, 0f x x
) 8f g x x 1xf x e
28. f 0,1 . :
) f x ) f x ) lnf x
29. f 5,8 .
g x f x f x
30. f(x) 1,4 , : ( ) ( 1) ( 3).h x f x f x
31. :f , : f x y f x f y ,
,x y :
) 0 0f ) f ) , ,f x y f x f y x y
) f
32. :f , : ( ) ( ) ( ) 1f xy f x f y xy
,x y . :
) ( )f ) 1 1
,( )
f xx f x
) ( )
1,( )
x f x xf x y
y f y y
33. :f :
) ( ) (2 23 )f x f x x ) 1 4
2 ( ) ,f x f xx x
.
34. :f :
) 2 ( ) 4 ( ) ,f x f x x x x ) 1 6
( ) 3 2 ,f x f x xx x
.
35. :f , : ( 1) ( )xf x e ef x ,
x .
) f ) f.
36. :f , 2 4( ) ,x f x x x x x
f.
37. :f , ln 1x
f x f xe
0x . ) f.
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14
5
) fC .
) f.
38.
) ( ) 2 lnf x x ) ( ) 1 2f x x x )2( ) 1x xf x e e )
1
xf x e
) 2( ) 2f x x x ) 2( ) 6 8f x x x )
3( )
2
xf x
x
)
2
2( )
1
xf x
x
39. .
) 2
2
3 , 0( )
2 , 0xx x x
f xx x
)
2 3 0( )
1xx x
f xe x
40. f, g R, (
).
) gof .
) fof gog.
) ,( 1)f x ln lnx x .
41. f . : ) f
) kf 0 k
0k .
) f k , k .
) ( ) 0f x 2f
42. :f : 1 2 1 21
( ) ( )2
f x f x x x
1 2,x x 1 2x x . :
) 2
xg x f x .
) 2
xg x f x .
43. f 0, :
f xf x e x 0,x . f .
44. f
: 22
f xe x x ,0x .
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14
6
45. : 0,f
: 2ln 4 1f x x x 2,x .
46. ) ( ) xf x x e (0,)
) , 0, :
e e .
47. ) ( ) ln xf x x e 0,
) x>3 : 3 ln
3
x xe e .
48. ( ) xf x x , x , 0,1 .
) f .
) 1,0 x 3 3x x x x
49. :f 3f x f x x x .
) f .
) 3 3 2f x f x
50. :f
3 3 22 2 5 1f x f x x x x x .
) f
) 3 1 1f x f x .
51. :f 3 3 3f x f x x
x .
) f
) 1 1f 3 0f . ) :
i. 2 2f f x f f x ii. 0f x iii. 1f x iv. 2 4 0f f x 52. 3( ) 5 1f x x x , x .
) f
) ( ) 7f f x .
53. ( ) lnxf x x , x >0 0,1 .
) f 0,
) : 2 2 2 24 9 ln( 4) ln( 9)x x x x x x
54. :f .
2 1fof x x fof x
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14
7
55. 2 2 lnf x x x .
)
) f .
) 0f x .
) 0f x .
56. f R ( ) ( ) ( )f x f y f xy x,yR.
(3 4) ( 2) (6 4) (2 1)f x f x f x f x .
57. f ( (3 7 0) )f x f x
x . A f
) f(x)=0 ) 2 0( )3f x
58. :
) 2( ) 2 3f x x x ) 2( ) 4 5f x x x
)6 3( ) 4 3f x x x ) 2( ) 2 2x xf x e e
59. : 2y x
.
60. 2( ) 2f x x x , , x . , f
2 .
1-1
1 - 1
61. 1-1 :
) 2
(x)1
xf
x
) (x) 2f x ) (x) x xf
) 2
2(x)
4
xf
x
)
2(x)
1
xf
x
)
2(x) xf e x
)
3
(x) xf e
) 2
(x) ln2
xf
x
62. 1-1 :
) ( ) 1f x x ) ( )f x x x ) ( ) 2
xf x e x
) ( )f x x )
2
3
2
xf x
x
1-1
63. ) A : f : g 1-1 ,
g f 1-1.
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14
8
) f 1-1 3( ) [ ( )] (2 3 )x f x f x 1-1
64. :f : ( ) ( ) 2f f f x f f x x
x .
i) f .
ii) fC .
65. :f : 3( ) 4 ( ) 2 3f x f x x
x . :
) f 1-1 , ) fC 1,1A .
1 - 1
66. A f : f(f(x))+f3(x)=2x+3 , x . ) f 1-1 ) f(2x3+x)=f(4-x)
67. f : R R f (f (x)) = f (x) + ex -1 x R.
) f 1-1.
) f(x3 5x) = f(2x -6).
68. f : R R 1-1 ( ( )) ( )f f x f lnx lnx 0x .
) f
1,0 .
) 3 2 0( ) f x x .
69. f : R R (1,1) (2,4).
) :
i) 2 2 4f x ii) 3 ( ) 1f x ) :
i) 2 1( )3f x ii) 1( )1 4f x
70. :
) 1 2 3f x x ) x -
f x- x
)
x
x
ef x
e -
) 1( ) ( )f x ln x
71. 1-1 .
)3( ) 2f x x )
1( ) 4xf x e ) 1
( )2
xf x
x
)2( ) 10 9, 6h x x x x )
3 3( )
3 5
xa x
x
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14
9
72. 3 5, 6
( )2 1, 6
x xf x
x x
) f
) f
.
73. f 4 1, 3
( )10, 3
x xf x
x x
.
) f 1-1
) f-1
) f
1f .
74. :f ( ) ( ) 0f x f x x , x :
) f ) f
75. :f : ( ) ( ) 2f f x f x x x
) f ) 1f .
76. :f :
4 15f f x x ( )( ) 8 35f f f x x x .
) f . ) f.
77. :f : ( ) ( ) 1f f x f x x x
. : ) f , ) fC 1,1A .
78. :f ( )( ) ( )f f x f x x , x :
) f ) fC .
79. :f : 1( ) 3 2 xf f x e
x . :
) f ) fC 1,1A .
) f .
--
80. :
) ln 1x x ) 2 2 2x xe x e ) 7 92 3 0x x
) 11 55 113 33 1 1 3x x x x x x
) 3 31
ln 7 1 7 11
xx x
x
ee e
e
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14
10
81. :f : ( ) 2 1f f x x
x .
) f .
) fC 1, 1A .
) ( )f x x .
82. :f : 3( ) ( ) 2 3f f x f x x ,
x .
) f ,
) 32 4f x x f x , x .
83. :f 2f xe f x x x . ) f
) lne
f x fx
) 1f
) 3 8 3 1xx e f .
84. 1
x
x
ef x
e
(x) lng x .
) f 1f .
) 1f og x .
) 1 e , : 1 ln ln
1 ln ln
.
85. f :
, , .
) ( )f f x x
) ( )f x
86. :Rf R
(-1,5) (2,4)
) f
) 1 2 3 5f f x
) 1 3 3 5xf f e
87. 2 2( ) 2 3 2 ,xf x e x e x .
) f
) 1 2( ( 2 ) 1) 3f f x e .
) 1 2( ( ) 1 2 ) 0f f x e .
88. 5 3( ) 3 2 1f x x x , x .
) f
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14
11
) 1 4 2 4f f x
) 1 2 2 2 5 0f f x x
89. :f : 5fof x x x . ) :
i. f ii. 55f x f x x .
) i. f x x .
ii. : 5 51 1 0f f f
iii. 32 243f , f .
fC 1fC f
90. ( ) xf x e x
) ( ) ( )f x f x
) ( )f x x .
91. ( ) xf x x
) ( ) ( )f x f x
) ( ( )) ( )xf f x f
) ( )f x .
92. ( ) 4f x x x .
) f f
) fC 1fC .
93. ( ) 3 9,xf x x x .
) f 1( 5)f .
) ( ) ( )f x f x
) 1( (ln ) 3) 0f f x .
94. 5( ) 4 128,f x x x x .
) f .
) 1 1( 128), (2)f f .
) 1,f fC C .
95. :f : 3( ) 3 ( ) 3f x f x x
x .
) f 1f .
) f .
) ( ) ( )f x f x .
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14
12
0
0
96. f .
:
) x 2lim ( )
f x )
x 2lim ( )
f x
) x 4lim ( )
f x )x 6lim ( )
f x
97. f .
:
) x 1lim ( )
f x )
x 1lim ( )
f x
) x -1lim ( )
f x )x 1lim ( )
f x
) x 3lim ( )
f x )
x 3lim ( )
f x
) x 5lim ( )
f x
98. :
) 3
21
2lim
5 4
x
x x
x x )
4
22
4lim
2
x
x
x )
4 2
31
5 4lim
1
x
x x
x )
3 3
2 2lim ,
x
x
x
99. :
) 21
1lim ,
1
v
x
xv
x )
3 2
31
1lim
2
x
x x x
x x )
4 2
4 3 22
2 3 2lim
5 14 80
x
x x x
x x x
100. : f , : 3 2( ) 3 ( ) 3 ( ) 9 f x f x f x x .
1
23
( )lim
5 6
x
f x
x x.
101. :
) 0
2 2lim
x
x x
x )
1
2 3 1lim
5 2
x
x
x )
3
3
5 2lim
3
x
x
x
-
14
13
) 3 3
5
2 3lim
5
x
x
x )
2
2
2 4lim
2
x
x x
x )
2
24
16lim
3 10
x
x
x x
)3
25
3 4 1lim
8 15
x
x x
x x )
3
21lim
1
x
x x
x )
3 4
1
2lim
1
x
x x
x
) 2
4
6 8lim
6
x
x x
x x x x )
4
4lim
2 9
x
x
x x
102. :
)
2
3
6lim
4 1
x
x x
x )
1
2 1lim
1
x
x
x )
2
3
1 4 9lim
3
x
x x
x
)
2
20
5 4lim
2
x
x x
x x )
21
1 2 1lim
x
x x
x x )
2
22
2 2lim
4
x
x x x
x
) 2
4
8 16lim
4
x
x x
x
103. :
) 2
1
| 3 4 | 2lim
1
x
x x
x )
3
31
| 4 1| 2lim
| 6 2 | 5
x
x x
x x
104. :
) 1
lim ( ),x
f x 2 1, 1
( )2 1, 1
x x xf x
x x
) 0
lim ( ),x
f x
2 1 2, 1 0
( )
6 1, 0
xx
f x x
x x
105.
23 5 2, 0
( ) | | 2
3 , 0
x xx
f x x
x
, 0
lim ( )x
f x ,
2
lim ( )x
f x .
106.
4
2
3 , 1
( ) 3 , 1 1
2 , 1
x x x
f x x x
x x
, f x1= 1
, x2=-3 x3=- 1.
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14
14
107. f 2 2
3 2
2 9, 1( )
5 2 2, 1
x a x xf x
x ax ax x.
1
lim ( )x
f x
108. f
2 1, 1
( ) 2 , 1 2
2 , 2
ax x x
f x ax x
a x
.
, R
1
lim ( )x
f x ,2
lim ( )x
f x .
109. :22 1 1
( )x
x x xf x
x
6 x 2( )
x 2
xg x
x.
, 1
lim ( )x
f x 2
lim ( )x
g x .
110. ,, 9 3 0 ,
2
3lim
3
x
x x
x.
111.
) 3
0lim
x
x
x )
2
20
1lim
x
x
x )
3
0lim
1 x
x
x
)20
2 2lim
x
x x
x )
20lim
x
x x
x )
2
30lim
x
x x x x
x x
112. :
) 20
3 2lim
x
x
x )
0
2lim x
x
x )
0lim
x
x
x
) 0
1 1lim
x
x
x )
0
1lim
2 1 1
x
x
x )
20
4lim
2
x
x
x x
113. :
) 0
1 1lim
x
x x
x )
0
4 4 3lim
x
x x x x
x x
) 0
3lim
3
x
x x
x x )
0
2lim
x
x
x
114. : f 2
2 4( )
4 12 32
x xf x
x x x
(3,4) (4,5)x . 4
lim ( )x
f x .
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14
15
115. x 4 8 4 8( ) x x g x x x , 0
( )limx
g x
x.
116. : f , 231 ( ) 1 1 x f x x x ,
x . 1
lim ( )x
f x .
117. : f , 2 2 ( ) 2 x f x x
0x . :
)2
lim ( )x
f x )2
( ) 4lim
2
x
f x
x )
22
( ) 5 3lim
4
x
f x
x )
22
( ) 3 1lim
5 6
x
f x
x x
118. 2 24 ( ) 3 2 x x f x x 3 x :
)1
lim ( )x
f x )1
( ) 5lim
1
x
f x
x )
1
( ) 1 4lim
3 2
x
f x
x )
1
( ) 4 3lim
2 ( ) 6 2
x
f x
f x
119. : f 22
( ) 5lim 3
5 6
x
f x x
x x.
2
lim ( ) 10
x
f x .
120. : f 2
0
( )lim 2
2
x
xf x x
x x.
0
lim ( ) 6
x
f x .
121. : f : 1
( )lim 3
1
x
f x x
x.
1
( 1) ( ) 2lim
1
x
x f x
x.
122. , : f g :
22
( )lim 3
3 2
x
f x
x x 2
2lim ( )( 4) 1
xg x x .
2lim ( ) ( )x
f x g x .
123. :
) 0
3lim
x
x
x )
20
2 1lim
x
x
x )
30
2 3lim
x
x x x
x
) 23
( 3)lim
7 12
x
x
x x )
0
4 5lim
2
x
x x x
x x.
124. ) 12
lim ( ) 3
x
f x , 2
lim (5 2)
x
f x .
) 0
( )lim 2016
x
f x
x,
0
(3 )limx
f x
x.
125. : f : ( ) ( 2) 2 2 f x f x x
x 2
lim ( ) 4 2 2
x
f x x . 0
lim ( )x
f x .
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14
16
126. : f 0
( )lim 3
x
f x
x ,
: 2 20
(4 ) ( ) 3lim
4
x
xf x f x x
x x
127. x R (5 2 ) ( ) f x f x 2
lim ( ) 2016
x
f x , 1
lim ( )x
f x .
128. : f : ( ) (2 ) f x f x
x 22
lim ( ) 2
xf x x x .
0lim ( )x
f x .
129. : f : 0
( )lim 2
x
f x
x.
2
20
(2 3 )lim
2 3
x
f x x
x x.
130. : f : 0
(3 )lim 4
( )
x
f x
f x.
0
(27 )lim
( )x
f x
f x.
131. : f , 0
2lim 0
x x
f x .
0
lim 0
x x
f x .
132. : f , 22
lim 4 ( ) ( ) 4
xf x f x .
2
lim ( )x
f x .
133. : f , 2 2( ) 2 ( ) 0 f x f x x
x . 0
lim ( ) 1
x
f x .
134. : f , 2 2( ) 4 ( ) 4 f x f x x ,
x . : 0
lim ( ) 2
x
f x .
135. , : f g
0
2 2lim 0
x xf x g x .
0 0
lim lim 0
x x x x
f x g x .
136. : 0, 0, f ,
2 2 23 ( ) ( ) x x f x xf x 0x .
0
( )limx
f x
x,
.
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14
17
0
k
137. f .
:
) x 2lim ( )
f x )
x 2lim ( )
f x
) x 4lim ( )
f x )
x 4lim ( )
f x
138. f .
:
) x 2lim ( )
f x )
x 2lim ( )
f x
) x 4lim ( )
f x )
x 4lim ( )
f x
139. , , :
) 3
2 7lim
3
x
x
x )
21
2 1lim
( 1)
x
x
x )
22
3 4lim
4 4
x
x
x x )
1
3 2lim
1
x
x
x
) 2
32
5 6lim
( 2)
x
x x
x )
25
3 1lim
25
x
x
x )
22
4 5lim
4
x
x
x )
2
2lim
2
x
x x
x
)2
21
3 4lim
1
x
x x
x )
3 23
4 7lim
6 9
x
x
x x x )
23
2 3lim
3 7 12
x
x
x x x
) 0
(3 )lim
x
x
x x )
0lim
x
x
x )
2
2lim
x
x
x
)0
2lim
1
x
x
x )
0
3lim
1
x
x
x )
0
2 7lim
3
x
x
x
140. :
) 2
1
2lim
1
x
x x
x )
22lim
( 2)
x
x
x.
141. 2
2
3 2( )
1
x xf x
x.
1lim ( )x
f x
, .
142. 2
2lim
2
x
x x
x ,
, .
-
14
18
143. 3 2
2
2 8( )
4
x x xf x
x, , ,
2lim ( ) 4
x
f x .
144. f: 2
2lim 2 ( ) 4
x
x f x .
: )2
lim ( )x
f x ) 2
22
3 ( ) 5 ( ) 7lim
( ) 2
x
f x f x
f x
145. 1
lim 4 ( ) 3 ( )
x
f x xg x 1
lim 2 ( ) 3 ( )
x
xf x g x
1
lim ( )x
f x 1
lim ( )x
g x .
146. f . f 0
lim
x
f x ,
0
lim
x
f x .
147. : f : 21
lim 2
x
f x x .
) 1
lim ( )x
f x
) 1
1lim ( )
( )
xf x
f x.
) g x f x x , 1
lim ( )x
g x .
148. : f :
8 2 2 24 f x y f x f y xy x y , x y .
) 3 , f x x x .
) : i. 201
limx f x
ii.
30limx
f x
f x
) f
1 1
0
1 1 2lim
x
f x f x
x
149. :
) 0
5lim
xx
x )
2
3
2 5lim ( 3 )
3
x
xx x
x
) 0
3 3lim
x
x
x x )
2
20
3lim 2
1
x
xx x
x x
) 1821
0
2lim
x
xx x
x )
200
21
2lim ( 1) 1
1
x
x xx
-
14
19
150. f .
:
) xlim ( )
f x ) xlim ( )
f x
) x
1lim
( ) f x )
x 0
1lim
( ) f x
151. f .
( ):
) xlim ( )
f x ) xlim ( )
f x
) x
1lim
( ) f x )
x 0
1lim
( ) f x
-
152. .
) 3lim -5 3 - 2
xx x ) 5lim - 2 3 -1
xx x
)4lim 2 - 1
xx n x ) 7lim - - 2
xx x
153. .
)
4
3
2 1lim
2 1x
x x
x x
)
3
3
2 3 1lim
5 1x
x x
x x
)
2
3
5 1lim
1x
x x
x x
)
2
3 3 1lim
1 3 2
x
x
x x x
154. .
)
2 5 , 1
( ) 7, 1
x x a x
f xa x
x
)
3
3
3
4 1, 5
2 3( )4
, 58 1
xx
xg xx
a xx
155. f f(x)= 22
, x - - (2, )4
- x
x
x .
:
) lim ( )x
f x ) lim ( )x
f x )2
lim ( )x
f x ) 2
lim ( )x
f x .
-
14
20
156. :
) 2lim 4 1 2 3
x
x x x ) 2lim 25 1 5
x
x x x
) 2lim 2 9 1
x
x x ) 2lim 2 3
x
x x
157. :
) 2
2
4 5 1lim
9 2 2 4
x
x x
x x )
2
3 5lim
1x
x
x x
) 2
2
1lim
4 1 2
x
x x
x x )
54 34
32 2
16 3 2lim
1 4
x
x x
x x
158. :
) 2 2 2lim 2 4 3 1 5 10
x
x x x x x x .
) 2 2lim 1 9 2 4 1
x
x x x x x ) 2 2 2lim 4 2 9 1 25 1
x
x x x x
159. :
i) 2
2
| 3 | 1lim
1
x
x x x
x x ii)
2| 2 | 3lim
| 1| 2
x
x x
x iii)
3
4
| 2 3 | 1lim
2 2
x
x x x
x x
160. :
) 3 2
2
(2 ) 3lim
3 5
x
x x
x x )
3
2
(2 3) 3 1lim
( 2) 2
x
x x
x
161. :
) 2lim ( 1) 3 4
x
x x ) 2 3 2lim ( 9) 4 1
x
x x x .
162. :
) lim
x
x x ) lim
x
x x ) 2
lim
x
x
x
)
2 1
lim2
x
xx
x )
2
4lim
2
x
x
x ) lim
1
x
x x
x
) lim 2
x
x x ) 2
lim2
x
x x
x )
2 2lim
1
x
x x
x
-
14
21
) 22 3
lim4
x
xx
x )
2
lim
x
x x x
x x )
2
1
lim
x
x
x
) 1
lim 1
x
xx
) 1
lim
x
xx
)
2
2
1
lim2
x
163. :
) 2
2
2 2lim
9
x
x x x
x x )
2
4
10 5 2lim
x
x x x x
x x )
2lim
1
x
xx
x
) 4 2
( 1) 2lim
3 2
x
x x
x x )
3
3lim
3 4
x
x x x x
x x )
25 2lim
4
x
x x x
x x
164.
) 2
20
3 4 3 3lim
3 2 3 3
x x
x xx )
2 1
20
2 2 1lim
3 4 2 1
x x
x xx )
1
1 3
2 3lim
2 3
x x
x xx
) 1
1
4 6 2lim
2
x x
x xx
e
e )
1
1
5 2lim
3 5 2
x x
x xx )
2 1
2
2 3lim
2
x x
x xx
e
e
)1
2
4 3 2lim
4 5 3 1
x x
x xx
165. 3 3 5
( )3 8
x x
x xf x , 0 .
: lim ( ) 64
x
f x .
166. :
)
22 3
1lim
x
x
xe )
2
1
5lim
x
x x
xe )
2 1
2lim
x x
x
xe )
2 3lim
x x
xe
)0
3lnlim
ln ln
x
x
x x ) 2lim ln ln 2 3
xx x x
) 2lim ln 3 2
x
xe x ) 2lim 2ln( 1) ln( 1)
x
x x x
167. ) x : 3 3( )2 3 2 2 x x f x x x , x xlim ( ), lim ( )
f x f x
) x 2| ( ) 1 | 1 xf x x ,
xlim ( )
f x
) i) A 2
xlim ( ) 0
f x , xlim ( ) 0
f x
ii) g 2
xlim [ ( ) 4 ( )] 4
g x g x .
xlim ( )
g x
-
14
22
168. ) lim ( )x
f x 2
2
2 1 2 1( )
1 1
x xf x
x x , x < -1
) lim ( )x
f x 2 24 1 ( ) 1 x x f x x x , x > 0
169. : f 2lim ( ) 2 1 4
x
f x x x x .
lim ( )x
f x lim ( )x
f x .
170. ( )
lim 3
x
f x
x lim ( ) 3 6
xf x x ,
2
2 3
( ) 3 6lim
2 ( ) 6 4 5
x
xf x x
x f x x x
171. : f : 2( 1) ( ) 1 x f x x
x . lim ( )x
f x .
172. x :
2lim
1
x
x
x
21
lim 31
x
x
x.
173. 2 2 2
2
2 2lim
1
y
y x xy x yf x
xy, 0x .
) 2
, 0
x x
f x xx
) limx
f x .
174. ln 1 ln 1 x xf x e e .
) ) f.
) f .
) :
i. 4 3 2
3 2
4 3 7 6 2lim
5 9
x
f x x x x
x x ii.
1
lim
f x f x
xe e
175. : f : 3 3 4 f x f x x
x .
) f .
) f .
) 2 42 3 x xf )
1
4lim
x
f x x
x.
-
14
23
176. f .
:
) f
) f
) 2 , 4f f )
.
177. f .
:
) f
) f
) 2 , 0f f )
.
f x0.
178. 0x x :
) , 0
( )1, 0
xe xg x
x x(
0x 0 ) )
2 3 2, 1
( ) 1
2, 1
xx
f x x
x
(0x 1 )
)
2 1, 1
1
( ) 2, 1
3 1 1
xx
x
f x x
x x
(0x 1 ) )
1 1, 0
( ) 0 , 0
,x2
xx
x
f x x
x
x
f Df
179. :
)
2| 16 |, 4
( ) 4
8 , 4
xx
f x x
x
)
2016 2
2016, 0
( )
0 , 0
xex x
f x x
x
.
-
14
24
180. N :
2
2
, 0
5 4, 0 1( )
4 3 , 1 2
3 4, 2
x x
x a xf x
x x a x
x x
181.
2 1
2
xxf x
x, 0x , 0 .
) limx
f x .
) lim 0
0 0
xf x
g .
182. ,, R f :
2
2
2
3 2, 1
1
3( ) , 1
1
, 1
x xx
x
x axf x x
x x
x
.
183. f
, 0
( ) , 0
,
x x
f x a x x
x x
.
, f R.
184. , x x -
, 2( ) x -
x , 2
xf x
x
185. : f 0 0x :
2
0
1( )
lim 2
x
xf x x
x. (0)f .
186. f x0 = 0 x R :
2 22 ( ) 2 x x xf x x x (0)f .
187. f 0 0x , : 2( ) xf x x x x
x . (0)f .
188. : f 0 2x 3( )( 2) 6 4 f x x x x
x . (2)f .
-
14
25
( )
189. : f 0 0x 2( ) ( 3) 2 6 19 f x f x x x
x (0) 5f . f 1 3x .
190. : f 0 0x :
0
( )lim 2
x
f x
x ( ) ( 1), f x f x x .
) (0)f . ) f 1 1x .
191. : f :0
( )lim 2
x
x xf x
x x. fC
0, 3A f 0 0x .
192. , : f g :
2 2 2( ) ( ) 2 ( ) ( ) f x g x f x g x x , x . g
0 0x ,
f 0 0x .
193. , : f g
2 2 2( ) 4 ( ) 4 ( ) ( ) f x g x f x g x x x . g
0 0x , f 0 0x .
194. : f : 3( ) ( ) 3 f x f x x
x . f .
195. f R f 5 (x) + f (x) +1 = ex. f x0 = 0 .
-
196. : f , 0 2x ,
1
1lim 3
1
x
f x
x, .
2
( ) (2)lim
2
x
f x f
x.
197. : f , 0 1x 0( 1)
lim 4
x
f x
x.
1
( ) (1)lim
1
x
f x f
x.
198. : f 0 6x :
3
( 3)lim 2
3
x
f x
x.
6
( ) (6)lim
6
x
f x f
x.
199. : f 0 0x 2( ) ( 1) 4 4 8 f x f x x x
x (0) 3f . f 1 1x .
200. f 2 f(3)=10 f(5-x)=f(x) x , : ) f(2)
-
14
26
) f x0=3
201. : f : ( ) x f x x
x .
) 3f ) f 0 x .
) 21/3
1lim
xx f
x
202. : f : ( ) ( ) 2 ( ) f x y f x f y
, x y . f 0 0x , f
.
203. : f : ( ) ( ) f x y f x f y
x, y . f 0 3x , f .
204. : f , 0 0x , :
( ) 2 ( ) f x y f x f y xy x, y . f .
205. : f 0 1x :
( ) ( ) ( ) f xy f x f y x, y . f .
206. : f ( ) ( ) 2016 , , f x f y x y x y ,
f 0.
olzano
207. 2 4 3 2
( )2 3 2
x x xf x
x x .
f Bolzano 0,4 0,4 , ( ) 0 f .
208. 2,0 , , : 4 33 2 5 0 .
209. 1
4 0 xex
0,2 .
210. f, g , : 2 2( ) ( ) 5 f x g x x
( ) ( )f x g x x . ( ) ( ) 0 f x g x
, 0 0 .
-
14
27
211. : 0, f : 22ln ( ) ln x x f x x x 0x .
2( ) 1 ln f x e x x 1,e .
212. : f : 22 ( ) xx f x e
x . 3( ) f x e x 0,1 .
213. f . f f x x
, f x x .
214. f 2 2 23 f f x f f x x
x 2 2 f .
) 0f x .
) 0 , f , 20 , x
, 0 0f x .
215. f 0,2 0 ( ) 1 f x 0,2x .
0,2 , 2 ( ) ( ) f f .
216. f, g 0,1
f g 0,1 ,
0 0,1x , : 2
0 0 02 1 f g x g f x x .
217. 2 2 , 0 x x
22 .
2
218. 5 24 3 0 x x 0, .
219. 2,1 :
) 3 23 1 0 x x ) 4 22 1 0 x x ) 6 3 22 5 1 0 x x x x
220. ,,, 0 , 0 0 .
x x x 0
, .
221. f : [2,6] R [2,6] . :
1 1 1
( ) 02 4 6
f xx x x
, (2,6).
222.
-
14
28
) 3 2 4 0 x x (1,2) ) 4 32 3 8 0 x x x ,
223. 2ln 3 0 x x 0, .
224. : 0,1 f (0) (1) 0 f f .
( ) 0f x 0,1 .
225. : , 0 f :
0 e f e f . ( ) 0f x
.
226. : 0,1f . 0 0,2
x
f x x .
227. : f (1) (5)f f .
1,3 2 f f .
228. : 0,2 0,1f . 0,2
2 2 2 3 f f .
229. : 1,2 0,1f . 0 1,2x
, : 2 0 0 0 01 f x x f x x .
230. f 0,1 0 1 f 1 4f . f y=x.
231. f, g 0, 2 0 0 , 2 2 f g f g .
, y f x y g x
0 0, 2 . x
232. f 0,1 ,
2 ( )3 3 x f x x x x 0,1 . f
4 1 y x 0,1 .
- f
233. f, g 2( ) 5 6 ( ) f x x x g x x . 2, 3 f (2) (3) 0g g .
234. : g : 0 g g g ,
g x x . g .
-
14
29
235. : 0, f :
2 4 f x xf x 0x 3 4 f .
) 2
x
g x f x ,
.
) f.
) f x k , k .
236. f , :
3 2 2( ) 5 ( ) ( ) 7 12 f x f x f x x x x . f
[-1,2].
237. f
2 4( ) 3 ( ) 2 xf x f x x e . f .
238. : f :
2 4 2( ) 2 4 ( ) 3 f x x x f x x . : ) (1)f ) f.
239. f : 2 2 xf x e
x 0 1f .
240. f
2 2 2 f x x f x (1) x .
241. f 3,3
2 2 ( ) 9 x f x 3,3 x .
) ( ) 0f x .
) f 3,3 .
) f (0) 3 f .
242. f 2 3 2 0 f x f x .
f .
243. 4( ) 3 4 f x x x , f 200 , 2,4x .
244. f , , .
,
2
3
f f f
f .
245. f 0,8 0,8 ,
0 1,4x ,
0
0 2 2 3 4 2 8
8
f f f ff x .
-
14
30
246. : 2,3 f , : (2) 2f , (3) 3f
2 2( ) 3 ( ) 2 f x f x x x 2,3x . f .
247. : f 1 3f 2 5f .
4f x , f .
f
248. f , . 1 2 3, , x x x , ,
, , , :
1 2 3( ) ( ) ( ) 4 ( ) f x f x f x f .
249. f 1,4 , 0 1,4x ,
0
1 3 2 5 3 4
10
f f f ff x .
250. f , ( ) 0f x , x .
1 2, ,..., , x x x , , , :
1 2( ) ( ) ( )... ( )
f f x f x f x .
- f- f
251. 2ln xx x e 0,
252. f lim
x
f x .
1
0 f xx
.
253. 1821 0 xe x .
254. ln 1 1 xf x e x .
) f ) ln 1 1 xe x
255. :[1,3]f . 3,5 , f .
256. 3 0 x x , 0 1 0
.
257. 1
ln
x
f x xx
, 0x .
) f )
) ln 1 x x x .
) , , 1 , 2 f f f .
258. : f ( ) 0f x x (3) 5f .
-
14
31
2(4) 3 3 1
lim5
x
f x x
x.
259. : f (3) 5 f 1, 4
( ) 0f x . 3
2
(0) 2 5lim
4
x
f x x
x.
260. 52 3( ) f x x x 0,2x . ) f .
) .
) 1( ) ( )f x f x .
261. f 0,2 . 2
( ) 1lim 16
( 2)
x
f x
x
f , :
) Cf Cg ,
3( 2) g x x x .
) 22
( ) 1lim
5 6
x
f x
x x .
262. ( ) ln 1 xg x x e x .
) .
) , (x) f .
) (x) f e .
) : 4 4 2 2ln(4 ) ln( 4) 4 4 e e .
263. : f , : 2 ( ) 4
1
x
x
f xe
e
x .
) f .
) f 0 , :
i) f.
ii) 2( ) 4
lim3 4
x
x xx
f x.
-
14
32
264. f ,
A 2,f (2) .
x x o120 . f (2) f(2) .
265. , f 1, 2
, .
:
f 3 f 4 f 4 f 3
f 3 f 4
.
266. f,g ,
fC 2,f 2 gC 2,g 2 .
) f 2 , g 2 .
) : 8f 2 g 2 g 2 f 2 .
0( )f x
267. 0x :
i)
2
2
3 4 , 2
( ) 34 , 2
8
x x
f xx x
0 2x ii) 2
2 , 0( )
2 , 0
x xf x
x x x
0 0x
iii)
33 5 , 1
( ) 1, 1
2
x x x
f xx
x
0 1x iv)
23 5, 1( ) x 1
6 2, 1
x xf x
x x
268. 0 0x .
i)
2 , x
x0 1
lnx
x x x
f xx
ii) 2 1 , x
0
xf x x
x
0f (x )
269. ,g : f , : 3( ) ( ) ( ) ( ) f x g x f x x e
-
14
33
x . f 0 x e ( ) 3 f e ,
( ) 3 g e .
270. f,g 0 0x :
1 f x g x x x . 0 0 f g , :
0 0 1 f g .
271. : f , 0 0x , :
( ) 3 f x x x x .
) 0 f . ) 0
(9 ) ( )lim 4
(2 )
x
f x f x
f x.
272. f, x0 = 0
3 2 2 4 3( ) ( ) ( ) x f x x f x x f x x xR .
0 0f
0 .f
x )f
273. : f 0 0x
3 2 2 3 22 ( ) 2 ( ) ( ) 2 f x x f x x f x x x x . (0) 1 f .
274. , : 0,5 f g , : ( ) 2 ( )g x xf x
0,5x . f,g 0 2x (2) 0 f , (2) 2 (2) g f .
275. : f 0 1x
3 2( ) 2 ( ) 3 2 f x f x x x . 1f .
276. 23 1 x 1
1
xf x
x x . ,
f 0x .
277. ,,
2 x 1
f x x x, 1
1
x x
xx
0 1x .
278. 22 3 4 , 0
( )2 , 0
x x xf x
x x, , , f
0 0x .
-
14
34
279. f 0 2x x :
2( ) ( 3 2) ( ) g x x x f x ,
g 0 2x (2) (2) g f .
280. , : f g , g 0 x
( ) f x x g x , x . f
0 x , , ( ) 0g x .
281. f f(l) = 1 f '(l) = 2 3
2
( ), 1( )
, 1
f x xg x
ax x.
, g 1.
282. : f 0 1x :
0
(1 ) 3lim 5
h
f h
h .
) f 0 1x .
) 2
21
2 ( ) 4lim
1
x
f x x x
x.
283. f 0 0x , :
20
2lim 1
x
f x x
x.
:
) 0 0f ) 0 f )
0
2lim 4
1
x
f x x x x
x
284. f 0 2x
0
(2 )lim 3
h
f h
h. :
) (2) 0f ) f 0 2x .
285. f 0 3 x 0
( 3)lim 2016
x
f x
x.
f 0 3 x ( 3) f .
286. : f 0 x . :
) 0 0 0
0
2lim 0
h
f x h f x h f x
h
) 0 0 0
0
2 2lim 0
h
f x h f x h f x
h
)
2 2
0 0
0 00
2lim 2
h
f x h f x hf x f x
h
)
0 0 00
lim 2
h
f x h f x hf x
h
287. f x0, :
-
14
35
) 0 00
lim ( 5 ) ( )
h
f x h f x
) 0 0 00
( 5 ) ( 5h)lim 10 ( )
h
f x h f xf x
h
) 3 3
20 00 0
0
( 5 ) ( )lim 15 ( ) ( )
h
f x h f xf x f x
h
288. : f , 0 4x , f 0 .
(3 1), 1
( )(6 2), 1
f x xg x
f x x
0 1x g 0 .
289. : f 0 1 :
f 0 f .
2 x
12 1
2
f x
g x
f x x
g
1
2 f 0 f .
290. : f : f x y f x f y xy
, x y . f 0 0x ,
.
291. f , 0 0x , (0) 5 f
( ) ( ) f x y f x y f y x , , x y . f
.
292. f , : f x y f x f y
0f x , x y . :
) 0 1f
) f 0 0x , .
) f 0 0x 0 1 f , f
f x f x x .
293. .
) 4 35) 2 1( f x x x x )
3 22 3( )
3 2
x xf x kx
) 3
( ) 22
xln
f x x x e ) ( ) 5 f x x lnx x
) 3( ) 3 tf t x t ) 3 f
-
14
36
294. :
) (x) x lnxf ) xf (x) x e x ) xf x t e x
) 2 2
1
xf x
x )
ln
xf x
x )
xf x
x
295. , , :
) 2 x
x 0
x xf x
x x )
2 1 x
0
xf x
x x
) 23 x 1
x 3 1 1
x xf x
x x )
3 , 0( ) x
0, 0
x xf x
x
296. :
) 2(x) (x x ) f )
3
(x) xf e ) (x) ln x f
) (x) ( x ) f ) (x) ( x) f )x(x) (e ) f
) (x) x x f ) (x) x f ) 2 5(x) 3 x xf
) 1 (2 3) f x x ) 3 1 f x x ) 2
1( )
1 x
f x
297. .
) 3 2( ) f x x ) 54( ) f x x
) ( ) , 0 xf x x x )
ln( ) , 0 xf x x x
) 2( ) (3 ) , 0 xf x x x ) ( ) 3 , 0 x xf x x
298. f : R R . g :
) 32 5 g x f x ) 23
g x f xx
f
) 3 ( ) ( ) g x f x ) 3( ) 2 ( ) g x ln f x
) ( ) f xg x f x e ) 3 2( ) ( ) xg x f e .
299. f 3 46 1 f x x
x . 1f 0f .
300. f,g: 0, :
( x)(x) gf e . :
03 0
0
fg
f.
301. :
) 4 3
2( ) 3 24 3
x x
f x x x ) ( ) lnxf x x
-
14
37
) 2( ) 3 f x x ) ( )
x
f xx
302. f(x) =2 , 0
( )2 , 0
x xf x
x x x . f
.
303. : f . :
) f , f .
) f , f .
304. ( )P x , 2
( ) 8 ( ) 1 P x P x , x .
305. 1 2( )
f x x x , , * ,1 2, .
: 1 2
( )
( )
f x
f x x x
1 x 2 x .
306. :
) 0
1lim
x
x
e
x )
2 2
1lim
1
x
x
e e
x )
0lim
xe
x
e e
x )
ln 1lim
x e
x
x e
-
307. f , :
( ) ( ) ( ) f x y f x f y , x y . ( ) ( ) 0 f x f y .
308. , : (0, ) f
( ) ( ) ( ) f xy f x f y , , 0, x y . ( ) ( ) 0 xf x yf y ,
, 0x y .
309. : f 5 3 2 f x f x x
x .
) f ) 1 (1) f .
310. 5 3( ) ln f x x x
) f f-1
) 1f , 1 (1) f .
311. 2 5( ) ln 4,x 1, xf x e x
) 1f
) 1f , 1 (5) f
312. ( ) 2 xf x e x , x .
-
14
38
) f ) 1 (3) f .
313. 3( 1) 6 f x x x , Cf
xx.
314. 2( ) (2 ) xf x ln e . Cf
0 l 3 y n .
315. 2
xf x x , x>0 . fC
0 1x
1
4E .
316. : f , 2( ) x xf x x e e ,
x . fC
1, (1)A f .
317. 2 3 2( ) f x x x .
Cf, :
) x'x = 135.
) :3 2 0 x y .
) xOy .
) ' .
318. Cf :
) y=2x, f(x)=2x+1 ln x
x
) x+3y+1=0, f(x)=3x-2
ln x
x.
319. f(x) = x3 +x2 +9x-2. , R Cf (2, -10) : 3 1 0. x y
320.
2 , 1
( ), 1
x ax x
f xx
x
. ,, R
( c ) f
1, 1 f , 3 3 0 x y .
321. 2 8
( )2
xf x
x
2( ) 4 g x x x . ,
fC 0, (0)M f gC .
-
14
39
322. Cf 3( ) 1 f x x o
( )0,2 .
323. (c) ( ) xf x e
(c) .
324. f x x x , , x , .
fC , 2 2 0 .
Cf
325. 2 2 3( ) f x x x . : 2 3 y x
Cf.
326. y x
3 2( ) 6 8 f x x x x . ,
fC .
327. 2( ) ln 1 f x x x , 0x . x x f.
328. 3 2( ) 5 1 f x x x x 2( ) 2 3 g x x x .
,f gC C , .
329. 2
( )2
x
f x
2 1
( )2
x xg x
x ,
.
330. 2 f x x x 2 2 3 g x x x . ,f gC C
.
331. ( ) lnf x x ( ) xg x e .
fC x x gC y y ,
,f gC C .
332. f, g f(x) = g2
1
x . : y = 2x
Cg x0 = - 1 Cf x1 = 1.
333. f R f '(2) = 1 g
-
14
40
g(x) = f(ex +1) - 2 , xR .
Cf (2, f (2)) Cg B(0,g(0)).
334. , : f g , g ,
: 1f x g x g x g x x .
) f .
) 1 2, f,g
,A k f k ,B k g k ,
, :
i. 1 2 ii. MN
Rolle
- . Rolle
335. Rolle ,
. , ,
( ) 0 f .
) 2( ) 3 f x x x , 0,3x ) 2( ) 6 8 f x x x , 2,4x
)2
3
4, 0( )
4, 0
x xf x
x x, 1,1 x )
2
2
8 , 2( )
10, 2
x x xf x
x x x, 1,3x
336. f 2,3 f f , :
) Rolle 2,3 ( )
( ) f x
g xx
.
) , , f f .
337. , , R 2
,0 1( )
20 9,1 2
x xf x
x x x
. Rolle [0,2].
338. ,
2
2
4 , 1,0( )
1, 0,1
x x xf x
x x x, , , .
, , , f . Rolle 1,1 .
339. 4 3 22 12 1 0 x x x x ,
.
Rolle
340. 3 2( ) 2 3 f x x x x , , , x .
, ( ) M f fC 0,1 , ,
fC , x x .
-
14
41
341. g 32 1 g x x x .
0,1 () C g
, g (0, g(0)) (1, g(1)). ().
342. : f 2 2
14
f x
f x
x . f
0,1 , .
- f . Rolle F
343. 2016 20152017 2016 1 0, x a x a a (0, 1).
344. , , , 03 2
,
3 2 0 x x x , 0,1 .
345. f, 2,2 .
2,2 , ,
) 8 f ) 2
2 f e ) 0 f
346. f : [, ] (, )
3 3( ) ( ) . f f
(, ) , 2( ) 3 . f
347. f,g [2,3] 2 3,) 3 2( g g
( ) (23 ) 1 f f . 2,3 (g ) 1 f g .
348. f 0,1 0,1
1 0 1. f f 0,1 2-
( ) .
f
349. : 0,3
f , : 0 2
3
f f .
0,3
, 0 f f .
350. f,g, , , ,
( )
) ( )
f f
g g. ( ) 0g x , x ( ) 0 g x , x ,
, , : ( )
) ( )
f f
g g.
-
14
42
eF(x)
351. f 0,1 0 1f ef .
2 0 f x xf x 0,1 .
352. f,g 3,4 3 4f f , 3g
2 2 0 x xe e 4g 3 0 x x .
0 f x g x f x 0,1 .
353. f .
2
2
1
xf xf x
x .
354. f [,], (,)
f e f , , . (, )
0 f f
355. f 0,1 1 0 1 f f .
) 1 0,1 , : 1 12 f .
) 2 0,1 , : 2
2
2
1 1
1
f ff .
( ) 0 f x
356. f, , , 0 ,
( ) ( ) (0) f f f . , , , ( ) 0 f .
357. :[2,6]f 3 , (2) (6)f f
(2) (6) 0 f f . , , : 3 ( ) 0 f .
358. f 0 1f , 1 1 f e
22 8 f e . 0,2 , 6 f e .
359. f 1 1f , 2 4 ln 2 f
2 1 f e e . 1, e , 2
12 0
f .
- -
360. f, , , , , ( ) 0 f x
, x . f .
361. 2 2 0 xe x x , 3 .
362. xxe e , * , x ,
-
14
43
.
363. : 2,3 f :
0 f x f x 2,3x . 0 f x
2,3 .
364. 1,2 .
) 3 2 3ln x x ) 4 3 230 20 x x x
365. :
) 3 0 x x x ) 3 23 15 2 0 x x xe e e ) 3 x xe x e .
366. f, 2,2 , :
26 3 12 xf x x f x , 2,2 x . :
) 2 2 0 f f ) f 2,2 .
367. f,g , :
0 f x g x f x g x x .
0f x 0f x .
368. : f . :
) ( ) 0f x ,
( ) 0 f x .
) ( ) 0 f x ,
( ) 0f x .
369. f, , : 3 0 4 f f
5 4 0f f . Cf xx.
370. f, 1,4 , :
1 2 3 4 f f f f . 1,4 , :
0 f .
371. : 0,3 0, f 0 1 2 3f f f f . f .
Cf
372. f, [, ], (, )
0 f f c [, ]. :
) ( )
( )
f xG x
x c, c [, ] Rolle , .
-
14
44
) c [, ], 0 , c Cf
0 (, )c f co (c, 0).
373. f, 1,3 , 1,3 , 3 (1) (3)f f .
1,3 , , fC
, ( ) M f .
374. 3 2( ) f x x x x , 0 ,
. :
) ( ) 0 f x 1 2, , 1 2 .
) 2 3
) 1 2 0 f f .
...
... f- f '-
375. ... .
) 2( ) f x x x , 1,2x ) ( ) xf x e , 0,1x
) 2
2
, 1( )
3 2, 1
x x xf x
x x x, 0,2x
376. ( )1
nxf x
x. :
) ... f 2[ , ]e e
) 2,( ) e e 0 f
) 1
f
377. f, 1,3 , (3) 12f (1) 4f . f,
4 2 y x .
378. f R , f f .
y = f(x) 3 0 x y .
379. f [0, 2], 0,2
2 0 4 f f . (0, 1) ,
)( 2 f .
380. f , f x 0 ,
, x . , :( )
( )( )( )
( )
f
ff ef
.
-
14
45
381. f 0,1 0 3f , 2 f x
0,1x . 2 3 f x x 0,1x .
382. f , f , f 1 f x
, x . f x x , x .
383. 2 xf x e x , x .
1 2, , x x , , , : 21 xf x e .
384. 2
2
2 1, 1 1( )
2,1 2
x xf x
x x x
) , R ... 1,2
) 1,2 3 2 1 f f f .
385.
2
( ) 1, 1
ax x,x
f xx
x
. ,
... f [0,2]
2 ) 0( 2 f x f f (0,2).
386. f [,] (,),
5( ) f x , x . 2 255 f
2 9 f , 5 3 .
387. f , 3 f 3 f .
1 2, , , : 1 2 6 f f .
388. f 0,9 , 0 9f f .
1 2 3, , 0,9 , : 1 2 3 0 f f f .
389. f 1,10 , 1 4f 10 9f .
1 2 3, , 1,10 , :
1 2 32 3 4 5 f f f .
390. f, , , , , , ,
5
f
5
f . 1 2, , ,
: 1 23 2 1 f f .
. Bolzano ,. ..
-
14
46
391. f 1,1 f 1,1 ,
1 1 f 1 5f . :
) 1,1 , 3 f .
) 1 2, 1,1 , 1 2
1 11
f f ) 0 3f .
392. f , 0 f x , x . :
) 0 , , :
0
2 3
5
f ff x .
) 1 2, , , : 1 0 2 02 3 f x f x .
) , , : 1 2
5 3 2
f f f.
393. f, 1,2 , 1 4f
2 1f . :
) 0 1,2x , , 0 03 2 f x x .
) 1 2, 1,2 , : 1 2 9 f f .
f"
394. f, 2,6 :
2 4 2 6 f f f . 0 2,6x ,
0 0 f x .
395. f, ,
1 2 0 3 f f f f . 0 f x , .
396. f 0,1 1 0 1 f f
0 1 1 f f . 0 f x
0,1 .
397. f .
fC , 0 f x .
398. f, ,
0 f x x . f, .
f, f"
399. f , 0 f
0 f f . , , 0 f .
-
14
47
400. f , 0 f f .
1 2, , 1 2 , : 1 2 0 f f .
401. , 0, , :
) 2 2
) 1 1 , 2
402. : ) 2 ln
e
e ) 2 ee e
403. ) 1 1 1
ln 11
x x x , 0x )
2 1 2ln
1 1 1
x
x x x, 1x
404. 1 1 x xx e xe , x .
405. :
) 3 37 3 33 6 4 ) 4 44 44 3 5 2
406. f f .
: 1 1 f x f x f x f x , x .
407. f , f .
2 1 1 f x f x f x x .
408. f , 0 f , , . f
, , ( ) ( ) 0 f f .
409. f, , :
2 4 f x x x , x . lim
x
f x .
410. f, , : 1 xf x e ,
x . lim
x
f x .
411. f f 5 0 f .
lim
x
f x .
412. f lim ( ) 0
x
f x .
lim 2 2 1 0
xf x f x f x .
413. f, 0 f x
x . fC 1,5 2,1 , :
) f 1-1.
-
14
48
) , 1 24 8 2 f f x . ) fC ,
: 3
24
y x .
414. f 3 f x f x x
0 1f . g x f x f x .
415. f 0, 2 f x xf x
0x . f 0, .
416. f 0 x xf x e f x e
x 0 1f . xf x e , x .
417. f, :
) 2 f x x x , 0 1f ) 2 3 f x x x , 0 1f
) 22 , 0 xf x e x x , 21 f e ) 2ln
x
f xx
, 0x , 1 0f
) 2
2
1
xf x
x, 0 0f ) 23 f x x x , 1
2
f
418. f 0,
) 1
ln
xf x e xx
1 1 f ) 2
x xxe ef x
x, 1 f e
) 2
ln 1
xf x
x, (1) 1f ) x xf x xe e , 0 1f
419. f 0,
) 6 f x x , 0 0 2 f f ) 212 9 f x x , 0 4 0 4 f f
420. f :
) 3 23 4 2 xf x x x x x 1 6f .
) 21 2 8 10 x f x x x x 0 2f .
421. f 0, :
) 23 1 f x xf x x 1 2f
) xxf x f x e 1 1 f e
) 2 2 xxf x x f x x e 1 1 f e
) 2 3 xf x f x x x x x 2
2 4
f
422. f :
) 1 0 xf x x f x 1 f e , 0x
-
14
49
) 1 xf x x f x 1 f e , 0x
423. f, , :
0 f x xf x f x x , 0 0 f 0 2f .
424. f, , :
2 0 f x f x f x , x 0 0 1 f f .
f
425. f, , x , :
) 5 36 4 2 f x xf x x x x ) 21 3 6 1 x f x x x f x
) 2 xf x xf x x e , 1 f e ) f x xf x , 1 1f
426. f, , :
) 32 xf x f x e , 0 2f ) 2 2 f x xf x x , 0 1 f e
) 2 4 1 f x f x , 0 1f ) 2 f x f x , 0 2f
427. f 2
3 1, 0
2 1, 0
x xf x
x x.
3
(1)2
f f.
428. f 2 6, 3
( )2 6, 3
x xf x
x x 1 20f .
f.
f
429. : f R R 0 0f , :
( ) f xe x , x
( )( )
( ) 1
f x
f x
xf x e
x e , x R . f.
430. : f R R 0 0f ,
2
( )
2 ( )( )3 ln3 2
3
xf x
f x
e xf x x
x , x R . f.
431. f :
)
xx ef x
f x 0 2f , 0f x
) 2 x f xf x e , 0 0f
)
3
2
4 2
3
x xf x
f x, (1) 2f , 1x , f 0x
432. f :
) 2 xf x f x e 0 1f
-
14
50
) 2 0 f x f x 0 0 0 f f
) 199 0 f x f x 0 0 0 f f .
433. f :
2 1 2 f x f x x xf x x 0 3f .
434. f :
f x f x f x x xf x 1 0f
435. f , :
2 1 f x y f x f y xy , x y 0 1 f .
436. f 0,
f xy xf y yf x , 0, x y 1 0f 1 1 f .
437. f 1, 1f e ln xf x f x x
1x .
) 0f x 1x ) f.
) 1 ln 1
, 1
438. f , :
1 f x f x x , 0 f 0 f .
) 1 xg x f x f x e .
) 1 2 xf x f x e x ) f.
) f .
439. , : f g f
22 f x xf x , 0f x x
(0) 1f
g x xf x
, x .
) g .
) :
f xx
) lim
x
xf x x .
-
14
51
1
440. :
) 4 22 5 f x x x ) 1 f x x ) 2 6 8 f x x x
) 2 3
2
xf x
x )
3 12 x xf x e ) lnf x x x
) 2
ln
xf x
x )
xef x
x )
ln
1
x xf x
x
441. :
) 22 1, 1
5 5, 1
x xf x
x x x )
2
3 2 , 2
2 1, 2
x xf x
x x x
) 2
3
4 4 , 0 4
3 , 0 4
x x xf x
x x x x ) 2 1 f x x x
f,f,f(3),
442. :
) 1 22 4 3 xf x e x x ) 22 ln 1 f x x x x
) 24 2 4 xf x e x x
443. f, 00, x , 0 0x ,
0 f x 00,x x 0 0f .
f x
h xx
00, x .
444. f f 0, . A 0 0f
2 ln2 x x
f xg x e
x, , 0, .
445. , :
) 3 2 3 2007 f x x x x .
) 2 2 2ln 1 f x x x , 0 .
f
446. :
) 22 1 ln 1 2 5 f x x x x x , 0x
) 21 xf x x x e , 0x
) 22 2 3 xf x e x x , 0x
) 6 23 2 1 xf x e x x x
-
14
52
447. 2 13( ) 2 xf x e , 1
2x
) f.
) .
) 1f .
448. 2 2 x xf x e e , x . ) f .
) 1
100
f
1
100
f .
) 10 5f , 2010f 3015f .
449. f , , :
0 f f 0 f x , x . 0f x
, x .
450. f , 0 f x
, x f f . 0 f f .
451. f , :
1 f x x f x , x . f .
452. :
) ln 1 2 x x , 2x ) 1 1 xe x , 0x . ) x x x , 0, x
) 2
xe 12
x
x , 0x ) 1
1
xxe x , 0x ) 1ln 1 xx e , 1x
) 2 4
12 24
x x
x , 0x ) 1 x x x , 0,2
x
453. , : , f g ,
f x g x , x f g .
f g , .
454. : 0, f 2 6 f x x
0x , 0 0 f 0 1f . : 2 31 1 x f x x
0x .
455. f 0,1 :
2 xf x e 0,1x 0 2 f .
) 0 1 0 f f
) f 0,1 .
-
14
53
.
456. :
) 22 ln 3 x x x ) 3 2 x xe e ) 2ln 1 6 0 x x x
) 4 4ln 1 x x ) 23 3 2 x x ) 18212ln 1 1 2 xx e x
457. ) 5 12
113 13
x x
f x .
) 5 12 13 x x x 2.
458. :
)22 3 3 0 x x l x ) , elnx x x e ) 1, 0 xe x x
459. : 22 3 7 2 1ln 3 7 2 ln 2 1 2 x x xx x x , 0x
460. ( ) xf x x e .
) f .
)
i) 3 2 3 x x xe e x x ii)
2 1 1 2 0 x x xe e x
) : 2 1 2 2 xe x
461. : 1, 0,1f e , 0 f x , 1,x e .
1 ln f x x x 1,e .
462. ) 3 2 1 2 f x x x x , x , .
) 3 2 1 2 x x x 0,1 .
463. f : 2
2 2
2 5
xf x
x x
x . f
2ln 2 5 2 g x x x , .
464. f, , :
3 2 3 22 6 1 f x f x f x x x x , x , 2 3 .
0f x , 0,1 .
465. f 1,3 . 1 3 f 4 f x
1,3x , 0f x 1,3 .
466. :
) 22 ln 3 x x x ) 3 2 x xe e ) 23 3 2 x x ) 3 4 5 x x x
-
14
54
467. :
) 3 22 7 4 0 x x x ) 4 3 24 4 2 0 x x x ) 3 22 3
2 16
x xx
468.
0, .
) 2 0 xxe ) 4 3 212 14 3 4 0 x x x
469. 3 3 1 0 x x .
470. f, 2 xf x e
x 0 1f . f x ,
.
471. f , . 0 f x ,
, x , f .
472. f , 0 4f 2 0f .
)
x
f xg x
e
( ) ( )f x f x , x .
) , : f f .
473. : f :
2 21 2 x f x x x 0x . 3
limx
f x
x.
474. f, , 0 f x
x . 2 2 A f f 1 1 B f f .
475. f, , ,
: 2 f x f x 0 f x , x . ) f .
) 0f x . )
f xg x
f x, x x , , 45o .
476. ;
) 3 22 3 12 6 f x x x x ) ln
x
f xx
-
14
55
) 2 2 3 f x x x ) 2 2 2 ,x [ , ] f x x x
) ln
ln
xf x
x x )
2 ln
x
f xx
.
477. :
) 21, 2
6 , 2
x xf x
x x x )
2, 0
5, 0
, 0
x x
f x x
x x
) 2
2
4 3, 1
6 6, 1
x x xf x
x x x
478. : 0, 0, g 0 g x
0x g e e .
ln
g xf x
g x.
479. f, , :
3 2 3 26 6 2 f x f x f x x x . 12 , x . ) f.
) f .
- f-
480. :
) 0 x x , 0,2
) 2 3 x x x , 0,
2
) ) 2 2ln 1 x x , 0x
) 2
2 1
xxx x , 0,2x ) 3 2 x3 6 6 6e x x x
481. 2
ln
xf x
x, 0x .
) f.
) 0x 2 2x ee x .
) 0 0x 2 2 x ex ,
e .
482. : 21
1 1e
xx x e 0,1x .
f(x) = 0.
483. f:RR f '(x) 0 xR. f
( ) ( ). xf e f x a
484. R , x3 -12x + = 0 .
xe xe x
-
14
56
485. x0 (0,l)
2( 3) 2 xf x e x x .
486. f (x) = 2x x - 2x - lnx. f
.
f
487. 1 xf x e x , x , 1 . ) f .
) .
488. 2 xf x x e , 0 , 0x . ,
f .
489. 3 22 6 18 3 f x x x x , x . x, f x, .
-
490. 4 3 2
20074 3 2
x x xf x x , , * , ,x . f
, 2 3 .
491. : f :
3 21 0 f x x f x . f.
492. 2 2 2
f x x x x ,
.
493. : f 0 f x
x . f , f .
494. f 0,2 0 2 1 f f f .
0 0,2x , 0 0 f x .
495. ) f 0,8 2 0 8 6 f f f f , f .
) f 0,8 8 2 6 0 f f f f , f .
496. f, 1,4
2 1 4 3 f f f f , f .
"1 1"
-
14
57
497. f 0,2 , 0 2 1 f f 1 4f .
0 1x 0 f x , :
) f 0,1 . ) f.
498. f , , 0 f f
0 2 f . 0 0x 0 f x :
) f 0, . ) f.
) 1 2, , , 1 2 1 f f .
) 1 , x , 122 2
f x f f .
499. f 1,5 1 3f , 5 4f
1,6 . :
) 1 2, 1,5x x 1 2x x
1 2 0 f x f x .
) 1,5 , 0 f .
) 0 1,5x , 3
0 0 0 0 f x f x f x x .
) 6 y x fC 1,5 .
Fermat
500.
i) 2 4 1, 1,2 ( ) f x x x x ii) ( 1)) , 0,2( xf x x e x
501.
3 2
2
2 15 24 19 , [0,1]( )
18 1 , (1,2]
x x x xf x
x x x
f
502. 3 2 2 f x x x x , , , x . , ,
f 1 1 x 2 1x .
503. , , 2( ) f x x x l x
x1 = 1 x2 = 3. .
504. f 0, :
3 21
24
f x x xf x x 0 . f x0,
x0.
-
14
58
Fermat
505. : g , :
2 1 xxg x e x x . 0 3g .
506. 1
2ln
x
xx
0x , 2 .
507. f, , : , 1 2( ) 2, xe f x x x x (1) 2 f . fC
(1,2)A .
508. : f , :
2 3 4 f x f f , x . :
) 3 4f f
) 3,4 ,
f , M f x x .
509. 4 3 2
20074 3 2
x x xf x x , , * , ,x .
f , 2 3 .
510. : f :
3 21 0 f x x f x . f.
511. 2 2 2
f x x x x ,
.
512. , f , R,
2 2( ) 1 2 ( ) f x x xf x f .
[, ]
513. : 1,1 f , 2 1 f x x f x
1,1 x . f .
514. : 0,1 f 0 0f . f 0
515. : 2,4 f ( ) 0, ( ) 0 f a f .
(,) ( ) 0 f
-
14
59
M -
516. f
f .
)
f
)
f
) f,
.
517.
f f..
)
f
)
f
) f,
.
518.
f f .
)
f
)
f
)
f .
519. ( ).
) 4 26 7 2 f x x x x ) 4f x x
) 2 1
2
x
x
ef x
e ) ( ) lnf x x x
-
14
60
520. :
) 2 3
2 3
2 , 0
4 , 0
x x xf x
x x x )
2
3 2
4 6, x
6 4 6, 0
x xf x
x x x x
521. 4
1 xf x e x , x .
) f .
) fC 0 0x .
) 4
1 3 2 xe x x x .
522. 1ln xf x x e , 0x .
) f 0, .
) 2 3 f x x 0x .
523. f :
1
3lim 4
1
x
f x x
x.
) fC 1, 1A f .
) A f , 7 4 0 f x x x .
524. || < 1, f(x) = 4 3 2
3 112 3 2
x ax x
x
R.
525. f (x) = x4 -x3 + 3
2 x2 -1
R.
526. 1
( ) ln , 0, 0 f x a x x ax
.
, Cf
4
,0
Ma
.
527. 3 22 1
2 53 2
f x x x x , x .
, f 03
2x .
528. 4 3 2 f x x x x x , 0 , x . fC
, 2 8
3 .
..
529. lnf x x x , 0x . ) f .
-
14
61
) 0 , 2
.
530. f, 0,5 2 0f .
3 0 2 5 0 f f .
531. f . :
1 1 f x f x f x f x , x .
f
532. f : R R ) ( ) ( )( 4 f x f x f x
x .
2( ) ( ), xg x e f x x , .
533. g, , :
2 24 0 g x x g x x x . g .
534. 3 2 2 23 2 2 2 2 x f x x x x . f
, .
535. 23 6 xf x x e , ,x .
fC , .
536. f 1 3 f x f x
x . f .
537. f f . f 1 0 , f .
538. f. A f
0x
f .
539. f . (1) 0 f ,
lim ( )
x
f x .
540. f ,
2
3lim 1
2
x
f x x
x 4 6f .
) 2 6f .
) fC 2, 2A f . ) 3 y x fC
-
14
62
0 2,4x .
) f , 2,4 f .
541. f 1,4 :
3 1 3
4
f ff x 2 f x 1,4x . :
) 1 3f f
) 3 0 f
) f 1,4 .
) 4 2 f .
542. f , :
3 2 1
,2 1
x
xf x x .
) g(x) f (x) x .
) f y =3x
.
) f .
) lim 1
xf x f x .
De L Hospital
0
0
543. :
) 2
0
2 2lim
4
x
x
e
x )
2
1
lnlim xx
x
e e )
22
ln( 2)lim
ln( )
xx
x
e e
544. :
) lim xx
x
e ) lim
xx
nx
e )
(1 )lim
x
x
n e
x
545. :
) 2lim
xx
e x x ) 2lim ln 2
x
x x ) lim ln 1
x
xe x
546. :
) lim ( )
x
x nx ) 2lim ( )
xx
x e ) 2lim ( 1)
xnx x x
) lim ( )
xx
e x ) 2 2lim ( 3 1)
x
xe x x ) 2lim 1 3
x
xx
) 2 2lim ( )
xx
x n x e ) lim (1 2 )
xx
x n e )2lim ( ln )
x
xe x
-
14
63
547. :
) 30
lim( )x
x nx ) 2lim ( )
x
xx e )
1lim 1
xx n
x
) 10
limx
x
nx
e
)
1
0lim ( )
x
xxe )
0limx
x nx
)
1
0
1lim ( 1) ln 1
x
xx e x
x )
1
lim ( 1)
x
xx e ) 3 2
0lim ln
xx x
0 0,1 ,
548. :
) 1
1lim 1
x
xx ,x>1 )
1
lim
x
xx )
1
lim ln
x
xx
) 0
lim
x
xx )
1
0lim 1
x
xx )
1
lim
x xx
e x
549. :
) 0
lim( 1)
xx
e nx ) 1
lim
xx
ex
) 0
lim(1 ) )
xx
e nx
De LHospital
550. 20
2 2 9lim
2
x
x
e x x x x
x, , .
551. ln 1 0 f x x x x x ) f .
) ln 1 x x x 0x .
) f.
552. f, .
)
20
2 2 2 ( )lim
4
h
f x h f x h f xf x
h
)
20
2 2 2lim 4
h
f x h f x f x hf x
h
553. : f lim 2
xf x f x .
limx
f x , :
) lim 2
x
f x ) lim 0
x
f x
554. : f lim 1
xxf x f x .
* lim
xf x , :
-
14
64
) lim 1
x
f x ) lim 0
x
f x
555. 2ln 2 ln 1 x x x 0, .
f
556. :
) 2
3
2x
xf
x x )
2x
5
xf
x )
2 3 4x
2
x xf
x
) 2x 4 3 1 2 f x x x ) ln
x x
fx
) x lnf x x
) 2
x1
x
x
ef
e ) x
x
xf ) x xf xe
557. 22 5 1
2
x xf x
x
y=2x-1.
558. 2
( ) 11
x
x
ef x x
e. :
) y = x + 1 Cf -.
) y = x - 1 Cf + .
559. f 0, . y x , 0 Cf , :
) lim
x
f x ) lim
xf x xf x
560. : f, g : RR g(x) = f(x) + x + x1
x x 0.
y = 2x + 3 Cf +,
Cg +.
561. : , : f g ( ) ( ,) xg x f xe x .
y = 2x +1 Cf x0=0, Cg +.
562. f ,
1 0f 2
l