γλώσσες
Σελίδες
Νομικός
14
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
2
1 , 5( )
ln( 1), 3
x xf x
x x x
. ,
f .
3.
x
2
e 1f(x)
3x 4 x 4. ,
f .
4. :
, 0 3
( ) 1, 1 0
2, 3 1
x x
f x x
x x
.
5. :
) ( )i f x x ) ( ) | |ii g x x ) ( ) | 1| 2iii h x x
,
6.
) 2 5 6
( )3
x xf x
x
) 2(x)= x +9- 6xf
) 13 3 3 2( ) x x xf x ) ( ) 1 2f x x
7. f , g g(x)=f 2(x)-2f(x)+x2+3 x , g .
8. f : : 3 2 2( ) 2 ( ) 5 ( ) x xf x f x f x e e , x . fC
xx.
9. f : R R 2
2
2 ( ) 3 ( ) 1( )
3 4
f x f xg x
x x
xR . Cg ' .
14
2
,
10.
2 2 4f x x x ,
2 2g x x x .
11. ( ) ( 2) ( 2)f x n x n x
x y=f(x)
ln5y .
12. f,g: , : xf(x) 1 g(x) e , x
f,g.
13. f(x) = x3+4x2+4x+1, g(x) = 2x2+5x +3. i) Cf Cg ; ii) 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. .
) 2f(x) ln(x 2) ln(2 x) g(x) ln(4 x )
)
x 1f(x) ln(x 1) ln(x 1) g(x) ln
x 1 ) 2f(x) 2ln(x 3) g x ln(x 3)
16. , ,
2x xf(x)
x 2
2x ( 1)x 2 3g(x)
x 1 .
17. f, g A :
f(x) f(x) g(x) 4 g(x) f(x) g(x) 4 8 x A . : f g .
18. 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
14
3
19. 2 9
( )2
xf x
x
2
1( )
6
xg x
x x
.
: f g f g ,f
g.
20. f (x) 1x g(x) 6 x .
f+ g ,f- g ,f g , ,f g
g f.
21. 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
22. 1, [ 2,0]
( )2, (0,2]
xf x
x
0, [ 3, 1]( )
3, ( 1,1]
xg x
x
.
: i) f g ii) f g
23. fog , gof :
) (x)=x+3,x [0,2] f , g(x)=-x+2 [-2,2]
) 2
( )2
xf x
x
, ( )g x x
) ( )f x x , 2g(x)= 9-x
)2, 1
( ), 1
x xf x
x x
, g(x)=4x 2
) ( ) 2 f x x , 2g(x)= x 1
24. x
xxf
3)(
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.
14
4
26. f :
) (f og)(x) = x2-4x , x R g(x) = x+4
) (f og)(x) = 51 x , x -1 g(x) = -x5
) (fog)(x) =1
1
x
x
, x > -1 g(x) = ( 1)n x
27. g :
) 2f g x x x 1f x x
) xf g x e x x 1f x x
) 2 1f g x x lnx, 0f x x
) 8f g x x 1xf x e
28. f 0,1 .
:
) f x ) 4f 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
.
14
5
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.
) 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, (
).
i) gof .
ii) fof gog.
iii) f (x) = ln [ln(x)], x > 1.
41. f . : i) -f
ii) kf k>0 k 0 f2
14
6
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 .
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 ln3
xe .
48. ( ) xf x x , x , 0,1 .
) f .
) 3 3x x x x 0,1x
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
14
7
) 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
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
14
8
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 )
3(x) 3x 2x 1f
) 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 )3( ) 2 3 2f x x x ) ( ) 2xf x e x
) 4( ) 5f x x ) 23
2
xf x
x
1-1
63. ) A f : g : 1-1 , g f 1-1.
) f 1-1 (x)=[f(x)]3+2f(x)-3 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 , ii) 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)
14
9
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 f (f (x)) = f (x) + ex -1
x R.
) f 1-1.
) f(x3 5x) = f(2x -6).
69. f : R R (1,1) (2,4).
) :
i) 2 2 4f x ii) f 3(x)=1 ) :
i) 2 1( )3f x ii) 1( )1 4f x
70. :
) f(x)= -1+2 3 x ) x-1
f(x)=2-x
) x
x
e +1f(x)=
e -1 ) f(x)=ln(1-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
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
f-1.
74. f : ( ) ( ) 0f x f x x , x :
) f ) f
14
10
75. :f : ( ) ( ) 2f f x f x x
x
) f ) 1f .
76. f : : f f x 4x 15
(f f f)(x) 8x 35 x .
) f . ) f.
77. :f : ( ) ( ) 1f f x f x x
x . : ) f , ) fC
1,1A .
78. :f ( )( ) ( ) 2f 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
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 .
14
11
83. f : f xe f x x 2 x .
) f )
ef lnx f
x
) 1f
) 3 xx 8 e 3 f 1 .
84. 1
x
x
ef x
e
(x) lng x .
) f f 1.
) 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
) 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 .
14
12
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 .
14
13
0
0
96. :
) 3
21
2lim
5 4x
x x
x x
)
4
22
4lim
2x
x
x
)
4 2
31
5 4lim
1x
x x
x
)
3 3
2 2lim , x
x
x
97. :f , : 3 2( ) 3 ( ) 3 ( ) 9f x f x f x x .
1
23
( )lim
5 6x
f x
x x
.
98. :
) 0
2 2limx
x x
x
)
1
2 3 1lim
5 2x
x
x
)
3
3
5 2lim
3x
x
x
) 3 3
5
2 3lim
5x
x
x
)
2
2
2 4lim
2x
x x
x
)
2
24
16lim
3 10x
x
x x
)3
25
3 4 1lim
8 15x
x x
x x
)
3
21lim
1x
x x
x
)
3 4
1
2lim
1x
x x
x
) 2
4
6 8lim
6x
x x
x x x x
)
4
4lim
2 9x
x
x x
99. :
)
2
3
6lim
4 1x
x x
x
)
1
2 1lim
1x
x
x
)
2
3
1 4 9lim
3x
x x
x
)
2
20
5 4lim
2x
x x
x x
)
2 1
1 2 1limx
x x
x x
)
2
22
2 2lim
4x
x x x
x
) 2
4
8 16lim
4x
x x
x
100. :
) 2
1
| 3 4 | 2lim
1x
x x
x
)
3
31
| 4 1| 2lim
| 6 2 | 5x
x x
x x
14
14
101. :
) 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
102.
23 5 2, 0
( ) | | 2
3 , 0
x xx
f x x
x
, 0
lim ( )x
f x
,
2
lim ( )x
f x
.
103.
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.
104. :
) 0
3limx
x
x
)
0
2limx
x
x )
0limx
x
x
) 0
1 1limx
x
x
)
20
2 1limx
x
x
)
20
4lim
2x
x
x x
105. :
) 30
2 3limx
x x x
x
)
0
4 5lim
2x
x x x
x x
)0
3 lim
3 x
x x
x x
)
23
( 3)lim
7 12x
x
x x
106. :f :
3 2( ) 3 ( ) 3 ( ) 1f x f x f x x x . 1
0
( ) 1limx
f x
x
.
14
15
107. :f 2
2 4( )
4 12 32
x xf x
x x x
(3,4) (4,5)x . 4
lim ( )x
f x
.
108. x 4 8 4 8( )x x g x x x , 0
( )limx
g x
x.
109. :f , 231 ( ) 1 1x f x x x ,
x . 1
lim ( )x
f x
.
110. :f , 2 2 ( ) 2x f x x
0x . :
)2
lim ( )x
f x
)2
( ) 4lim
2x
f x
x
)
22
( ) 5 3lim
4x
f x
x
)
22
( ) 3 1lim
5 6x
f x
x x
111. 2 24 ( ) 3 2x x f x x 3x :
)1
lim ( )x
f x
)1
( ) 5lim
1x
f x
x
)
1
( ) 1 4lim
3 2x
f x
x
)
1
( ) 4 3lim
2 ( ) 6 2x
f x
f x
112. 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
113. f
2,2
21,2
1,1
)(
2
xa
xax
xxax
xf
.
, R
)(lim1
xfx
, )(lim2
xfx
.
114. :22 1, 1
( ), x 1
x x xf x
x
6 , x 2( )
, x 2
xg x
x
.
, 1
lim ( )x
f x
2
lim ( )x
g x
.
14
16
115. ,, 9 3 0 ,
2
3
lim
3x
x x
x
.
116. :f 22
( ) 5lim 3
5 6x
f x x
x x
.
2
lim ( ) 10x
f x
.
117. :f 2
0
( )lim 2
2x
xf x x
x x
.
0
lim ( ) 6x
f x
.
118. :f : 1
( )lim 3
1x
f x x
x
.
1
( 1) ( ) 2lim
1x
x f x
x
.
119. , :f g :
22
( )lim 3
3 2x
f x
x x
2
2lim ( )( 4) 1x
g x x
. 2lim ( ) ( )x f x g x .
120. :f : ( ) ( 2) 2 2f x f x x
x 2
lim ( ) 4 2 2x
f x x
. 0
lim ( )x
f x
.
121. :f 0
( )lim 3x
f x
x ,
: 2 20
(4 ) ( ) 3lim
4x
xf x f x x
x x
122. :f : ( ) (2 )f x f x
x 22
lim ( ) 2x
f x x x
. 0lim ( )x
f x
.
123. :f : 0
( )lim 2x
f x
x .
2
20
(2 3 )lim
2 3x
f x x
x x
.
124. :f : 0
(3 )lim 4
( )x
f x
f x .
0
(27 )lim
( )x
f x
f x.
14
17
125. :f , 0
2lim 0x x
f x
.
0
lim 0x x
f x
.
126. :f , 22
lim 4 ( ) ( ) 4x
f x f x
.
2
lim ( )x
f x
.
127. :f , 2 2( ) 2 ( ) 0f x f x x
x . 0
lim ( ) 1x
f x
.
128. :f , 2 2( ) 4 ( ) 4f x f x x ,
x . : 0
lim ( ) 2x
f x
.
129. , :f g
0
2 2lim 0x x
f x g x
. 0 0
lim lim 0x x x x
f x g x
.
130. : 0, 0,f ,
2 2 23 ( ) ( )x x f x xf x 0x . 0
( )limx
f x
x,
.
131. :f f xe f x x
x .
) f .
) f 1f .
) f
1f .
)
2
1
1lim
1f xx
f x
x e
,
1lim 1x
f x
.
132. :f 3 2f x f x x
x .
) f .
) f 1f .
) 1
21lim
2x
f x
x x
)
2limx
f x
.
14
18
0
k
133. , , :
) 3
2 7lim
3x
x
x
)
21
2 1lim
( 1)x
x
x
)
22
3 4lim
4 4x
x
x x
)
1
3 2lim
1x
x
x
) 2
32
5 6lim
( 2)x
x x
x
)
25
3 1lim
25x
x
x
)
22
4 5lim
4x
x
x
)
2
2lim
2x
x x
x
)2
21
3 4lim
1x
x x
x
)
3 23
4 7lim
6 9x
x
x x x
)
23
2 3lim
3 7 12x
x
x x x
) 0
(3 )limx
x
x x
)
0limx
x
x
)
2
2limx
x
x
)0
2lim
1x
x
x
)
0
3lim
1x
x
x
)
0
2 7lim
3x
x
x
134. :
) 2
1
2lim
1x
x x
x
)
22lim
( 2)x
x
x
.
135. 2
2
3 2( )
1
x xf x
x
.
1lim ( )x
f x
, .
136. 2
2lim
2x
x x
x
,
, .
137. 3 2
2
2 8( )
4
x x xf x
x
, , ,
2lim ( ) 4x
f x
.
138. f: 2
2lim 2 ( ) 4x
x f x
.
: )2
lim ( )x
f x
) 2
22
3 ( ) 5 ( ) 7lim
( ) 2x
f x f x
f x
139. 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
.
14
19
140. f . f 0
limx
f x
,
0
limx
f x
.
141. :f : 21
lim 2x
f x x
.
) 1
lim ( )x
f x
) 1
1lim ( )
( )xf x
f x
.
) g x f x x , 1
lim ( )x
g x
.
142. :f :
8 2 2 24f 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 2limx
f x f x
x
143. :
) 0
5limx
xx
) 23
2 5lim ( 3 )
3x
xx x
x
) 2
0
9 3 3lim x
x
x x
) 220
3lim 2
1x
xx x
x x
) 18210
2limx
xx x
x
) 200
21
2lim ( 1) 1
1xx x
x
-
144. .
) 3lim ( -5 3 - 2)
xx x
)
5lim ( - 2 3 -1)x
x x
)4lim ( 2 -5 +1)
xx n x
)
7lim ( 2- - 2)x
x x
145. .
) 4
3
2 1lim
2 1x
x x
x x
)
3
3
2 3 1lim
5 1x
x x
x x
14
20
)2
3
5 1lim
1x
x x
x x
)
2
3 3 1lim
1 3 2x
x
x x x
146. .
)
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
147. f f(x)= 22
, x (- ,-2) (2, )4
3 , -2 x 2
x
x
.
:
) lim ( )x
f x
) lim ( )x
f x
)2
lim ( )x
f x
) 2
lim ( )x
f x
.
148. :
) 2lim 4 1 2 3x
x x x
) 2lim 25 1 5x
x x x
) 2lim 2 9 1x
x x
) 2lim 2 3x
x x
149. :
) 2
2
4 5 1lim
9 2 2 4x
x x
x x
)
2
3 5lim
1x
x
x x
) 2
2
1lim
4 1 2x
x x
x x
)
54 34
32 2
16 3 2lim
1 4x
x x
x x
150. :
) 2 2 2lim 2 4 3 1 5 10x
x x x x x x
.
) 2 2lim 1 9 2 4 1x
x x x x x
) 2 2 2lim 4 2 9 1 25 1x
x x x x
151. :
i) 2
2
| 3 | 1lim
1x
x x x
x x
ii)
2| 2 | 3lim
| 1| 2x
x x
x
iii)3
4
| 2 3 | 1lim
2 2x
x x x
x x
14
21
152. :
) 3 2
2
(2 ) 3lim
3 5x
x x
x x
)
3
2
(2 3) 3 1lim
( 2) 2x
x x
x
153. :
) 2lim ( 1) 3 4x
x x
) 2 3 2lim ( 9) 4 1
xx x x
.
154. :
) limx
x x
) limx
x x
) 2
limx
x
x
)
2 1
lim2x
xx
x
)
2
4lim
2x
x
x
) lim
1x
x x
x
) lim 2x
x x
) 2
lim2x
x x
x
)
2 2lim
1x
x x
x
) 22 3
lim4x
xx
x
)
2
limx
x x x
x x
)
2
1
limx
x
x
) 1
lim 1x
xx
) 1
limx
xx
)
2
2
1
lim2x
155. :
) 2
2
2 2lim
9x
x x x
x x
)
2
4
10 5 2limx
x x x x
x x
)
2lim
1x
xx
x
) 4 2
( 1) 2lim
3 2x
x x
x x
)
3
3lim
3 4x
x x x x
x x
)
25 2lim
4x
x x x
x x
156.
) 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
14
22
)1
2
4 3 2lim
4 5 3 1
x x
x xx
157. 3 3 5
( )3 8
x x
x xf x
, 0 .
: lim ( ) 64x
f x
.
158. :
)
22 3
1limx
x
xe
)
2
1
5lim
x
x x
xe
)
2 1
2limx x
x
xe
)
2 3lim x xx
e
)3 2
0
3ln +2lim
4ln +ln +1x
x
x x ) 2lim ln ln 2 3
xx x x
) 2lim ln 3 2xx
e x
) 2lim 2ln( 1) ln( 1)x
x x x
159. ) x : 3 3( )2 3 2 2x x f x x x ,
x xlim ( ), lim ( )f x f x
) x 2| ( ) 1 | 1xf x x , xlim ( )f x
) i) A 2
xlim ( ) 0f x
, xlim ( ) 0f x
ii) g 2
xlim[ ( ) 4 ( )] 4g x g x
.
xlim ( )g x
160. ) 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 ( ) 1x x f x x x , x > 0
161. :f 2lim ( ) 2 1 4x
f x x x x
.
lim ( )x
f x
lim ( )x
f x
.
162. ( )
lim 3x
f x
x lim ( ) 3 6
xf x x
,
2
2 3
( ) 3 6lim
2 ( ) 6 4 5x
xf x x
x f x x x
163. :f : 2( 1) ( ) 1x f x x
x . lim ( )x
f x
.
14
23
164. x :
2lim
1x
x
x
21lim 3
1x
x
x
.
165. 2 2 2
2
2 2lim
1y
y x xy x yf x
xy
, 0x .
) 2
, 0x x
f x xx
) lim
xf x
.
166. ln 1 ln 1x xf x e e . ) ) f.
) f .
) :
i. 4 3 2
3 2
4 3 7 6 2lim
5 9x
f x x x x
x x
ii.
1
limf x f x
xe e
167. :f : 3 3 4f x f x x
.
) f .
) f .
) 2 42 3x xf )
1
4limx
f x x
x
.
f x0.
168. 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
, x 0
2
xx
x
f x x
x
x
14
24
f Df
169. :
)
2| 16 |, 4
( ) 4
8 , 4
xx
f x x
x
)
2016 2
2016 , 0
( )
0 , 0
xex x
f x x
x
.
170. N :
2,43
21,34
10,45
0,
)(2
2
xx
xaxx
xax
xx
xf
171.
2 1
2
xxf x
x
, 0x , 0 .
) limx
f x
.
) lim , 0
0 , 0
xf x
g
.
172. ,, R f :2
2
2
3 2, 1
1
3( ) , 1
1
, 1
x xx
x
x axf x x
x x
x
.
173. f
, 0
( ) , 0
,
x x
f x a x x
x x
.
, f R.
174. ,
2x +2x-6, 2
( ) x-2
x+3 , 2
xf x
x
14
25
175. :f 0 0x :
2
0
1( )
lim 2x
xf x x
x
. (0)f .
176. f x0 = 0 x R :
2 22 ( ) 2x x xf x x x f(0).
177. f 0 0x , :
2( )xf x x x x
x . (0)f .
178. :f 0 2x 3( )( 2) 6 4f x x x x
x . (2)f .
( )
179. :f 0 0x 2( ) ( 3) 2 6 19f x f x x x
x (0) 5f . f 1 3x .
180. :f 0 0x :
0
( )lim 2x
f x
x ( ) ( 1), f x f x x .
) (0)f . ) f 1 1x .
181. :f :0
( )lim 2x
x xf x
x x
.
fC
0, 3A f 0 0x .
182. , :f g :
2 2 2( ) ( ) 2 ( ) ( )f x g x f x g x x , x . g 0 0x ,
f 0 0x .
183. , :f g
2 2 2( ) 4 ( ) 4 ( ) ( ) f x g x f x g x x x . g
0 0x , f 0 0x .
184. :f : 3( ) ( ) 3f x f x x
x . f .
185. f R f 5 (x) + f (x) +1 = ex. f x0 = 0 .
14
26
-
186. :f , 0 2x ,
1
1lim 3
1x
f x
x
, .
2
( ) (2)lim
2x
f x f
x
.
187. :f , 0 1x 0
( 1)lim 4x
f x
x
.
1
( ) (1)lim
1x
f x f
x
.
188. :f 0 6x :
3
( 3)lim 2
3x
f x
x
.
6
( ) (6)lim
6x
f x f
x
.
189. :f 0 0x 2( ) ( 1) 4 4 8f x f x x x
x (0) 3f . f 1 1x .
190. f 2 f(3)=10 f(5-x)=f(x) x , : ) f(2)
) f x0=3
191. :f : 26 ( ) 9x f x x
x .
) 3f ) f 0 3x .
) 21/3
1lim x
x fx
192. :f : ( ) ( ) 2 ( )f x y f x f y
,x y . f 0 0x , f
.
193. :f : ( ) ( )f x y f x f y
x, y . f 0 3x , f
.
194. :f , 0 0x , :
( ) 2 ( )f x y f x f y xy x, y . f .
195. :f 0 1x :
( ) ( ) ( )f xy f x f y x, y . f .
14
27
196. :f ( ) ( ) 2016 , ,f x f y x y x y ,
f 0.
olzano
197. 2 4 3, 2
( )2 6, 2
x x xf x
x x
.
f Bolzano 0,4 0,4 , ( ) 0f .
198. 2,0 , , : 4 33 2 5 0 .
199. 1
4 0xex 0,2 .
200. f, g , :
2 2( ) ( ) 5f x g x x ( ) ( )f x g x x .
( ) ( ) 0f x g x ,
0 0 .
201. : 0,f : 22ln ( ) lnx x f x x x 0x .
2( ) 1 lnf x e x x 1,e .
202. :f : 22 ( ) xx f x e
x . 3( )f x e x
0,1 .
203. f .
f f x x , f x x .
204. f
2 2 23f f x f f x x x 2 2f .
) 0f x .
) 0 , f , 20 ,x
, 0 0f x .
14
28
205. f 0,2 0 ( ) 1f x 0,2x .
0,2 , 2( ) ( )f f .
206. f, g
0,1 . f g 0,1 ,
0 0,1x , :
20 0 02 1f g x g f x x .
207. 2 2 , 0x x
22 .
2
208. 5 24 3 0x x
0,3 .
209. 2,1 :
) 3 23 1 0x x ) 4 22 1 0x x ) 6 3 22 5 1 0x x x x
210. ,,, 0 , 0 0 .
3 2x x x 0
1,1 .
211. f : [2,6] R [2,6] . :
1 1 1
( ) 02 4 6
f xx x x
, (2,6).
212.
) 3 2 4 0x x (1,2) ) 4 32 3 8 0x x x 1,3
213. 2ln 3 0x x (0,1).
214. : 0,1f (0) (1) 0f f .
( ) 0f x 0,1 .
215. : , , , 0f :
0e f e f . ( ) 0f x
, .
14
29
216. : 0,1f . 0 0,2
x
0 0 f x x .
217. :f (1) (5)f f .
1,3 2f f .
218. : 0,2 0,1f . 0,2
2 2 2 3f f .
219. : 1,2 0,1f . 0 1,2x
, : 2 0 0 0 01f x x f x x .
220. f [0,1] f(0) = -1 f(1) =4 . f y=x.
221. f, g [0, 2] f(0) < g(0), f(2) > g(2). y=f(x), y = g(x)
x0 (0, 2 ).
222. f [0,1],
2 ( )3 3x f x x x x [0, 1]. f
y = 4x -1 (0,1) .
- f
223. f, g 2( ) 5 6 ( )f x x x g x
2,3x . 2, 3 f (2) (3) 0g g .
224. :g : 0g g g , , , g( ) 0x x . g
.
225. : 0,f :
2 4f x xf x 0x 3 4f .
) 2
xg x f x ,
.
) f.
) f x k , k .
14
30
226. f 1,2 :
3 2 2( ) 5 ( ) ( ) 7 12f x f x f x x x x . f
[-1,2].
227. f
2 4( ) 3 ( ) 2xf x f x x e . f .
228. :f :
2 4 2( ) 2 4 ( ) 3f x x x f x x . : ) (1)f )
f.
229. f : 2 2xf x e
x 0 1f .
230. f
2 2 2f x x f x (1) x .
231. f 3,3
2 2( ) 9x f x 3,3x . ) ( ) 0f x .
) f 3,3 .
) f (0) 3f .
232. f
2 3 2 0f x f x .
f .
233. 4( ) 3 4f x x x , f 200 , 2,4x .
234. f , , .
,
2
3
f f f
f
.
235. f 0,8 0,8 ,
0 1,4x ,
0
0 2 2 3 4 2 8
8
f f f ff x
.
236. : 2,3f , : (2) 2f , (3) 3f
2 2( ) 3 ( ) 2f x f x x x 2,3x . f .
14
31
237. :f 1 3f 2 5f .
4f x , f .
f
238. f , . 1 2 3, ,x x x , ,
4 , , , :
1 2 3( ) ( ) ( ) 4 ( )f x f x f x f .
239. f 1,4 , 0 1,4x ,
0
1 3 2 5 3 4
10
f f f ff x
.
240. f , ( ) 0f x ,x .
1 2, ,..., ,x x x , , , :
1 2( ) ( ) ( )... ( )f f x f x f x
.
- f- f
241. 2ln xx x e 0,
242. f limx
f x
.
1
0f xx
.
243. 1821 0xe x .
244. ln 1 1xf x e x .
) f ) ln 1 1xe x
245. :[1,3]f . 3,5 , f .
246. x3 + x + = 0, >0 + + 1
14
32
248. :f ( ) 0f x x (3) 5f .
2(4) 3 3 1
lim5x
f x x
x
.
249. :f (3) 5f 1, 4
( ) 0f x . 3
2
(0) 2 5lim
4x
f x x
x
.
250. f(x) = 2x5+x + 3 x [0,2]. ) f .
) .
) f(x) = f-1(x) .
251. f [0,2]. 2
( ) 1lim 16
( 2)x
f x
x
f , :
) Cf Cg ,
3( 2)g x x x .
) 22
( ) 1lim
5 6x
f x
x x
.
252. ( ) ln 1xg x x e x .
) .
) , (x)f .
) (x)f e .
) :
2 4 4 2 2ln(4 ) ln( 4) 4 4e e .
253. :f , : 2 ( ) 4
1
x
x
f xe
e
x .
) f .
) f 0 3 , :
i) f.
ii) 2( ) 4
lim3 4
x
x xx
f x
.
14
33
0( )f x
254. 0x :
i)
2
2
3 4 , 2
( ) 34 , 2
8
x x
f xx x
0 2x ii) 22 , 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)
2
0
3 5, 1( ) x 1
6 2, 1
x xf x
x x
255. 0 0x .
i)
2 , x 0
x ,0 1
lnx
x x x
f xx
ii) 2 1 , x 0
0 , 0
xf x x
x
0
f (x )
256. ,g :f , :
3( ) ( ) ( ) ( )f x g x f x x e x . f 0x e
( ) 3f e , ( ) 3g e .
257. f,g 0 0x :
1f x g x x x . 0 0 1f g , :
0 0 1f g .
258. :f , 0 0x , :
2( ) 3f x x x x .
) 0 3f . ) 0
(9 ) ( )lim 4
(2 )x
f x f x
f x
.
259. f, x0 = 0
3 2 2 4 3( ) ( ) ( ) xf x x f x x f x x xR .
0 0f
0 .f
14
34
0(x )f
260. :f 0 0x
3 2 2 3 22 ( ) 2 ( ) ( ) 2f x x f x x f x x x x . (0) 1f .
261. , : 0,5f g , : ( ) 2 ( )g x xf x
0,5x . f,g 0 2x (2) 0f , (2) 2 (2)g f .
262. :f 0 1x
3 2( ) 2 ( ) 3 2f x f x x x . 1f .
263. 23 1, x 1
, 1
xf x
x x
. ,
f 0x 1 .
264. ,,
2
2
x, 1
f(x) x x+1, 1
1
x x
xx
0 1x .
265. 22 3 4 , 0
( )2 , 0
x x xf x
x x
, , , f
0 0x .
266. f 0 2x x :2( ) ( 3 2) ( )g x x x f x ,
g 0 2x (2) (2)g f .
267. , :f g , g 0x
( ) ( )f x x g x , x . f
0x , , ( ) 0g x .
268. f f(l) = 1 f '(l) = 2 3
2
( ), 1( )
, 1
f x xg x
ax x
.
, g 1.
14
35
269. :f 0 1x :
0
(1 ) 3lim 5h
f h
h
.
) f 0 1x .
) 2
21
2 ( ) 4lim
1x
f x x x
x
.
270. f 0 0x , :
20
2lim 1x
f x x
x
.
:
) 0 0f ) 0 2f )
0
2lim 4
1x
f x x x x
x
271. f 0 2x 0
(2 )lim 3h
f h
h
. :
) (2) 0f ) f 0 2x .
272. f 0 3x 0
( 3)lim 2016x
f x
x
.
f 0 3x ( 3)f .
273. :f 0x . :
) 0 0 0
0
2lim 0h
f x h f x h f x
h
) 0 0 0
0
2 2lim 0h
f x h f x h f x
h
)
2 2
0 0
0 00
2lim 2h
f x h f x hf x f x
h
)
0 0 00
lim 2h
f x h f x hf x
h
274. f x0, :
) 0 0
0lim ( 5 ) ( )h
f x h f x
) 0 00
0
( 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
14
36
275. :f , 0 4x , f 4 0 .
(3 1), 1
( )(6 2), 1
f x xg x
f x x
0 1x g 1 0 .
276. :f 0 1 :
f 0 f 1 .
12 , x
2
12 1 ,
2
f x
g x
f x x
g
1
2 f 0 f 1 .
277. :f : f x y f x f y xy
,x y . f 0 0x ,
.
278. :f , 0 0x , (0) 5f
( ) ( ) + ( )f x y f x y f y x , ,x y . f
.
279. :f , : f x y f x f y
0f x ,x y . :
) 0 1f
) f 0 0x , .
) f 0 0x 0 1f , f
f x f x x .
280. .
) 4 3 5( ) 2 1f x x x x ) 3 22 3
( )3 2
x xf x kx
) 3
( ) 22
x lnf x x x e ) ( ) 5f x x lnx x
) 3( ) 3tf t x t ) 3 f
14
37
281. :
) 2(x) x lnxf ) 3 xf (x) x e x ) xf x t e x
) 2 2
1
xf x
x
)
ln xf x
x )
xf x
x
282. , , :
) 2
3
,x 0
x , 0
x xf x
x x
)
2 1 ,x 0
, 0
xf x
x x
) 2
3
3 , x 1
2x 3 1 , 1
x xf x
x x
)
3 1 , 0( ) x
0, 0
x xf x
x
283. :
) 2 4(x) (x 3x )f ) 3
(x) xf e ) 2(x) 2ln xf
) (x) (3x 2)f ) (x) ( x)f ) x(x) (e )f
) 3 2(x) 2x xf ) 3(x) 4 xf ) 2 5(x) 3x xf
) 1 (2 3)f x x ) 3 1f x x ) 2
1( )
1 xf x
284. .
) 3 2( )f x x ) 54( )f x x
) ( ) , 0xf x x x ) ln( ) , 0xf x x x
) 2( ) (3 ) , 0xf x x x ) ( ) 3 , 0x xf x x
285. f : R R .
g :
) 3 2 5g x f x ) 2 3
fx
g x f x
) 3 ( ) ( )g x f x ) 3( ) 2 ( )g x ln f x
) ( ) f xg x f x e ) 3 2( ) ( )xg x f e .
286. f 3 46 1f x x
x . 1f 0f .
287. f,g: 0, :
(3x)(x) gf e . :
03 0
0
fg
f
.
14
38
288. :
) 4 3
2( ) 3 24 3
x xf x x x ) ( ) ln xf x x
) 2( ) 3f x x ) ( )x
f xx
289. f(x) =2 , 0
( )2 , 0
x xf x
x x x
. f
.
290. f : R R . : ) f , f ' .
) f , f ' .
291. ( )P x , 2
( ) 8 ( ) 1P x P x ,
x .
292. 1 2( )f x x x
, , * , 1 2, .
: 1 2
( )
( )
f x
f x x x
1x 2x .
293. :
) 0
1lim
x
x
e
x
)
2 2
1lim
1
x
x
e e
x
)
0lim
xe
x
e e
x
)
ln 1limx e
x
x e
-
294. :f , :
( ) ( ) ( )f x y f x f y , ,x y . ( ) ( ) 0f x f y .
295. , : (0, )f
( ) ( ) ( )f xy f x f y , , 0,x y . ( ) ( ) 0xf x yf y ,
, 0x y .
296. :f 5 3 2f x f x x
x .
) f ) 1 (1)f .
14
39
297. ( )f x x , 0,x . f
1( )f x .
298. ( )f x x , 0,x . f
1( )f x .
299. ( ) 2xf x e x , x .
) f ) 1 (3)f .
300. 3 1( ) 6f x x x , Cf
xx.
301. 2( ) (2 )xf x ln e . Cf
0 l 3y n .
302. 2
xf x x , x>0 . fC
0 1x 1
4E .
303. :f , 2( ) x xf x x e e ,
x . fC
1, (1)A f .
304. 2( 2) 3f x x x .
Cf, :
) x'x = 135.
) :3 2 0x y .
) xOy .
) ' .
305. Cf :
) y=2x, f(x)=2x+1 ln x
x
) x+3y+1=0, f(x)=3x-2
ln x
x.
14
40
306. f(x) = x3 +x2 +9x-2. , R Cf (2, -10)
: 3 1 0.x y
307.
1,
1,)(
2
xx
xaxxxf
. ,, R
( c ) f
(1,f(1)) , 3x-y+3=0.
308. 2 8
( )2
xf x
x
2( ) 4g x x x . ,
fC 0, (0)M f gC .
309. Cf 3( ) 1f x x
o ( )0,2 .
310. (c) ( ) xf x e
(c) .
311. 2( ) +f x x x , , , ,x , .
fC ,
2 2 0 .
Cf
312. 2( 3) 2f x x x . : 2 1y x
Cf.
313. y x
3 2( ) 6 8f x x x x . ,
fC .
314. 2( ) ln 1f x x x , 0x . x x f.
315. 3 2( ) 5 1f x x x x 2( ) 2 3g x x x .
,f gC C , .
14
41
316. 2
( )2
xf x
2 1
( )2
x xg x
x
,
.
317. 2f x x x 2 2 3g x x x . ,f gC C .
318. ( ) lnf x x ( ) xg x e .
fC x x gC y y ,
,f gC C .
319. f, g f(x) = g2
1
x
. : y = 2x
Cg x0 = - 1 Cf x1 = 1.
320. f R f '(2) = 1 g
g(x) = f(ex +1) - 2 , xR .
Cf (2, f (2)) Cg B(0,g(0)).
321. , :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
322. Rolle ,
. , ,
( ) 0f .
) 2( ) 3f x x x , 0,3x ) 2( ) 6 8f x x x , 2,4x
)
2
3
4, 0( )
4, 0
x xf x
x x
, 1,1x )
2
2
8 , 2( )
10, 2
x x xf x
x x x
, 1,3x
14
42
323. f 2,3 3 2 2 3f f , :
) Rolle 2,3 ( )
( )f x
g xx
.
) 2,3 , f f .
324. , , R 2
x ,0 x 1f (x)
x 20x 9,1 x 2
. Rolle [0,2].
325. ,
2
2
4 , 1,0( )
1, 0,1
x x xf x
x x x
, , , .
, , , f . Rolle 1,1 .
326. 4 3 22 12 1 0 ,x x x x ,
.
Rolle
327. 3 2( ) 2 3f x x x x , , , x .
, ( )M f fC 0,1 ,
, fC , x x .
328. g g(x) = 2x3 - x - 1 . (0,1) () C
g (, g())
(0, g(0)) (1, g(1)). ().
329. :f
2 2
14
f xf x x .
f
0,1 , .
- f . Rolle F
330. , , ,
04 3 2
,
3 2 0x x x , 0,1 .
14
43
331. f, 2,2 .
2,2 , ,
) 8f ) 2
2f e ) 0f
332. :[ , ]f , ,
0f f . , , ,
) 0f f ) 1 0f f ,
) 0f f ) 0f f
333. : 0,3
f
, :
0 23
f f
. 0,3
,
0f f .
334. f,g, , , ,
( )( ) ( )
f f
g g
. ( ) 0g x ,x ( ) 0g x ,x ,
, , : ( )( ) ( )
f f
g g
.
eF(x)
335. : 1,1 0,f 1 1f f .
1,1 , :
) 22f f ) 38f f ) 56 ff e
) 2
2
3f
f
) 32f f )
2ff e
f
336. f, , , 0 , ,
( ) ( ) 0f f . , , , :
) ( ) ( ) 0f f ) 2 0f f
) ( ) 2 ( ) 0f f ) 2 1 0f f
337. f .
2
2
1
xf xf x
x
.
14
44
338. f [,], (,)
f e f
, , . (, )
0f f
339. f 0,1 1 0 1f f .
) 1 0,1 , : 1 12f .
) 2 0,1 , : 2
2
2
1 1
1
f ff
.
( ) 0f x
340. f, , , 0 ,
( ) ( ) (0)f f f . , , , ( ) 0f .
341. :[2,6]f 3 2,6
(2) (6)f f (2) (6) 0f f . 2, 6 ,
: 3
( ) 0f .
342. f 0 1f , 1 1f e
22 8f e . 0,2 , 6f e .
343. f 1 1f , 2 4 ln 2f
2 1f e e . 1,e ,
21
2 0f
.
- -
344. f, , , , , ( ) 0f x
,x . f .
345. 2 2 0xe x x , 3 .
346. xxe e , * , x , .
347. : 2,3f :
0f x f x 2,3x . 0f x
2,3 .
14
45
348. 1,2 .
) 3 2 3lnx x ) 4 3 230 20x x x
349. :
) 3 0x x x ) 3 23 15 2 0x x xe e e ) 3x xe x e
350. f, 2,2 , :
26 3 12xf x x f x , 2,2x . :
) 2 2 0f f ) f 2,2 .
351. f,g , :
0f x g x f x g x x .
0f x
0f x .
352. :f . :
) ( ) 0f x ,
( ) 0f x .
) ( ) 0f x ,
( ) 0f x .
353. f, , :
3 0 4f f 5 4 0f f . Cf xx.
354. f, 1,4 , :
1 2 3 4f f f f . 1,4 , :
0f .
355. : 0,3 0,f 0 1 2 3f f f f . f .
Cf
356. f, [, ], (, )
0f f c [, ]. :
) ( )
( )f x
G xx c
, c [, ] Rolle
, .
ii) c [, ], 0 , c Cf
0 (, )c f co (c, 0).
14
46
357. f, 1,3 , 1,3 , 3 (1) (3)f f .
1,3 , , fC
, ( )M f .
358. 3 2( )f x x x x , 0 ,
. :
) ( ) 0f x 1 2, , 1 2 .
) 2 3
) 1 2 0f f .
...
... f- f '-
359. ... .
) 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
360. ( )1
nxf x
x
. :
) ... f [e, e2]
) (e, e2) f '() < 0
) f() >1
361. f, 1,3 , (3) 12f (1) 4f . f,
4 2y x .
362. f R f() = , f() = < . y = f(x) x - y + 3 = 0.
363. f [0, 2], (0,2) f(2) = f(0) +4.
(0, 1) , f )=2
364. f , f x 0 ,
,x . , :( )
( )( )( )
( )
f
ff ef
.
14
47
365. f 0,1 0 3f , 1 5f 2f x
0,1x . 2 3f x x 0,1x .
366. f , f , f 1f x
,x . f x x ,x .
367. 2xf x e x , x .
1 2, ,x x , , , : 21xf x e .
368.
2
2
2x 1, 1 x 1f (x)
x x 2,1 x 2
) , R ... [-1,2]
) (-1,2) 3f '() = f (2) - f (-1)
369.
2 1
( ) 1, 1
ax x,x
f xx
x
. ,
... f [0,2]
( )2 2 0f x f f (0,2).
370. f [,] (,),
5( )f x , x . 2 5 25f
2 9f , = 5 = -3.
371. f , 3f 3f .
1 2, , , : 1 2 6f f .
372. f 0,9 , 0 9f f .
1 2 3, , 0,9 , : 1 2 3 0f f f .
373. f 1,10 , 1 4f 10 9f .
1 2 3, , 1,10 , :
1 2 32 3 4 5f f f .
14
48
374. f, , , , , , ,
5
f
5
f
. 1 2, , ,
: 1 23 2 1f f .
. Bolzano ,. ..
375. f 1,1 f 1,1 ,
1 1f 1 5f . :
) 1,1 , 3f .
) 1 2, 1,1 , 1 2
1 11
f f
) 0 3f .
376. f , 0f x ,x . :
) 0 , , :
0
2 3
5
f ff x
.
) 1 2, , , : 1 0 2 02 3f x f x .
) , , : 1 2
5 3 2
f f f
.
377. f, 1,2 , 1 4f
2 1f . :
) 0 1,2x , , 0 03 2f x x .
) 1 2, 1,2 , : 1 2 9f f .
f"
378. f, 2,6 :
2 4 2 6f f f . 0 2,6x ,
0 0f x .
379. f, ,
1 2 0 3f f f f . 0f , .
380. f 0,1 1 0 1f f
0 1 1f f . 0f x
0,1 .
14
49
381. f .
fC , 0f x
.
382. f, ,
0f x x . f, .
f, f"
383. f , 0f
0f f . , , 0f .
384. f , 0f f .
1 2, , 1 2 , : 1 2 0f f .
385. , 0, , :
) 2 2
) 1 1 , 2
386. : ) 2 lne
e
) 2ee e
387. ) 1 1 1
ln 11x x x
, 0x )
2 1 2ln
1 1 1
x
x x x
, 1x
388. 1 1x xx e xe , x .
389. :
) 3 37 3 33 6 4 ) 4 44 44 3 5 2
390. f f .
: 1 1f x f x f x f x , x .
391. f , f .
2 1 1f x f x f x x .
392. f , 0f , , .
f , ,
( ) ( ) 0f f .
14
50
393. f, , :
2 4f x x x , x . limx
f x
.
394. f, , : 1xf x e ,
x . limx
f x
.
395. f f 5 0f .
limx
f x
.
396. f lim ( ) 0x
f x
.
lim 2 2 1 0x
f x f x f x
.
397. f, 0f x
x . fC 1,5 2,1 , :
) f 1-1.
) , 1 24 8 2f f x . )
fC ,
: 3
24
y x .
398. f 3f x f x x
0 1f . g x f x f x .
399. f 0, 2f x xf x
0x . f 0, .
400. f 0x xf x e f x e
x 0 1f . xf x e , x .
401. f, :
) 2f x x x , 0 1f ) 2 3f x x x , 0 1f
) 22 , 0xf x e x x , 21f e ) 2ln x
f xx
, 0x , 1 0f
14
51
) 22
1
xf x
x
, 0 0f ) 23f x x x , 1
2f
402. f 0,
) 1
lnxf x e xx
1 1f ) 2
x xxe ef x
x
, 1f e
) 2ln 1x
f xx
, (1) 1f ) x xf x xe e , 0 1f
403. f 0,
) 6f x x , 0 0 2f f ) 212 9f x x , 0 4 0 4f f
404. f :
) 3 23 4 2xf x x x x x 1 6f .
) 21 2 8 10x f x x x x 0 2f .
405. f 0, :
) 23 1f x xf x x 1 2f
) xxf x f x e 1 1f e
) 2 2 xxf x x f x x e 1 1f e
) 2 3xf x f x x x x x 2
2 4f
406. f :
) 1 0xf x x f x 1f e , 0x
) 1xf x x f x 1f e , 0x
407. f, , :
0f x xf x f x x , 0 0f 0 2f .
408. f, , :
2 0f x f x f x , x 0 0 1f f .
f
409. f, , x , :
) 5 36 4 2f x xf x x x x ) 21 3 6 1x f x x x f x
) 2 xf x xf x x e , 1f e ) f x xf x , 1 1f
410. f, , :
) 32 xf x f x e , 0 2f ) 2 2f x xf x x , 0 1f e
14
52
) 2 4 1f x f x , 0 1f ) 2f x f x , 0 2f
f
411. f :
)
xx ef x
f x
0 2f , 0f x
)
32 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
412. f :
) 2xf x f x e 0 1f
) 2 1f x f x x 0 1f
) 2 0f x f x 0 0 0f f
) 199 0f x f x 0 0 0f f .
413. f :
2 1 2f x f x x xf x x 0 3f .
414. f :
f x f x f x x xf x 1 0f
415. f , :
2 1f x y f x f y xy ,x y 0 1f .
416. f 0,
f xy xf y yf x , 0,x y 1 0f 1 1f .
417. f 1, 1f e lnxf x f x x
1x .
) 0f x 1x ) f.
) 1 ln 1
, 1
14
53
418. f , :
1 , f x f x x , 0 2f 0 1f .
) 1 xg x f x f x e .
) 1 2 , xf x f x e x ) f.
) f .
419. , :f g f
22f x xf x , 0f x x
(0) 1f
21 g x xf x
, x .
) g .
) : 21
1f x
x
)
lim 2x
xf x x
.
14
54
1
420. :
) 4 22 5f x x x ) 1f x x ) 2 6 8f x x x
) 2 3
2
xf x
x
)
3 12x xf x e ) lnf x x x
) 2ln x
f xx
) xe
f xx
) ln
1
x xf x
x
421. :
) 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 1f x x x
f,f,f(3),
422. :
) 1 22 4 3xf x e x x ) 22 ln 1f x x x x
) 24 2 4xf x e x x
423. f, 00, x , 0 0x ,
0f x 00,x x 0 0f .
f x
h xx
00, x .
424. f f 0, . A
0 0f 2 ln 2x x
f xg x e
x ,
, 0, .
425. , :
) 3 2 3 2007f x x x x .
) 2 2 2ln 1f x x x , 0 .
f
426. :
) 22 1 ln 1 2 5f x x x x x , 0x
14
55
) 21 xf x x x e , 0x
) 22 2 3xf x e x x , 0x
) 6 23 2 1xf x e x x x
427. 2 13 2( ) xf x e , 1
x2
) f.
) .
) 1f .
428. 2 2x xf x e e , x .
) f .
) 1
100f
1
100f
.
) 10 5f , 20 10f 30 15f .
429. f , , :
0f f 0f x ,x . 0f x
,x .
430. f , 0f x
,x f f . 0f f .
431. f , :
1f x x f x , x . f .
432. :
) ln 1 2x x , 2x ) 1 1xe x , 0x . ) x x x , 0,x
) 2
xe 12
xx , 0x )
11
xxe x
, 0x ) 1ln 1xx e , 1x
) 2 4
12 24
x xx , 0x ) 1x x x , 0,
2x
433. , : ,f g ,
f x g x ,x f g . f g
, .
14
56
434. : 0,f 2 6f x x
0x , 0 0f 0 1f . : 2 31 1x f x x
0x .
435. f 0,1 :
2xf x e 0,1x 0 2f .
) 0 1 0f f
) f 0,1 .
.
436. :
) 22 ln 3x x x ) 3 2x xe e ) 2ln 1 6 0x x x
) 4 4ln 1x x ) 23 3 2x x ) 18212ln 1 1 2xx e x
437. ) 5 12
113 13
x x
f x
.
) 5 12 13x x x 2.
438. :
) 22 3 3 0x x l x ) ,elnx x x e ) 1, 0xe x x
439. : 22 3 7 2 1ln 3 7 2 ln 2 1 2x x xx x x , 0x
440. f(x) = x + ex
) f .
)
) 3 2 3x x xe e x x )
2 1 1 2 0x x xe e x
) : 2 1 2 2xe x
441. : 1, 0,1f e , 0f x ,
1,x e . 1 lnf x x x
1,e .
442. ) 3 2 1 2f x x x x , x ,
.
) 3 2 1 2x x x 0,1 .
14
57
443. f :
22 2
2 5
xf x
x x
x . f
2ln 2 5 2g x x x , .
444. f, , :
3 2 3 22 6 1f x f x f x x x x , x , 2 3 .
0f x , 0,1 .
445. f 1,3 . 1 3f 4f x
1,3x , 0f x 1,3 .
446. :
) 22 ln 3x x x ) 3 2x xe e ) 23 3 2x x ) 3 4 5x x x
447. :
) 3 22 7 4 0x x x ) 4 3 24 4 2 0x x x ) 3 22 3
2 16
x xx
448.
0, .
) 2 0xxe ) 4 3 212 14 3 4 0x x x
449. 3 3 1 0x x .
450. f, 2xf x e
x 0 1f . f x ,
.
451. f , . 0f x ,
,x , f .
452. f , 0 4f 2 0f .
) x
f xg x
e
( ) ( )f x f x , x .
) , : f f .
14
58
453. :f :
2 21 2x f x x x 0x .
3limx
f x
x.
454. f, ,
0f x x . 2 2A f f
1 1B f f .
455. f, , ,
: 2f x f x 0f x , x .
) f .
) 0f x .
)
f xg x
f x
, x x , ,
45o .
456. ;
) 3 22 3 12 6f x x x x ) ln x
f xx
) 2 2 3f x x x ) 2 2 2 ,x [ , ]f x x x
) ln
ln
xf x
x x
)
2 ln
xf x
x .
457. :
) 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
458. : 0, 0,g 0g x
0x g e e .
ln g xf x
g x .
14
59
459. f, , :
3 2 3 26 6 2f x f x f x x x . 12 , x .
) f.
) f .
- f-
460. :
) 0x x , 0,2
) 2 3x x x , 0,2
) ) 2 2ln 1x x , 0x
) 2
2 1xxx x
, 0,2x ) 3 2 x3 6 6 6ex x x
461. 2ln x
f xx
, 0x .
) f.
) 0x 2 2x ee x .
) 0 0x 2 2x ex ,
e .
462. : 21
1 1e
xx x e 0,1x .
f(x) = 0.
463. f:RR f '(x) 0 xR. f
( ) ( ).xf e f x a
464. R , x3 -12x + = 0
.
465. x0 (0,l)
2 3( 2) xf x e x x .
466. f (x) = 2x x - 2x - lnx. f
.
f
467. 1xf x e x , x , 1 .
) f .
xe xe x
14
60
) .
468. 2xf x x e , 0 , 0x . ,
f .
469. 3 22 6 18 3f x x x x , x . x, f x, .
-
470. 4 3 2
20074 3 2
x x xf x x
, , * , ,x .
f , 2 3 .
471. :f :
3 21 0f x x f x . f.
472. 2 2 2
f x x x x ,
.
473. :f 0f x
x . f , f .
474. f 0,2 0 2 1f f f .
0 0,2x , 0 0f x .
475. ) f 0,8
2 0 8 6f f f f , f .
) f 0,8 8 2 6 0f f f f ,
f .
476. f, 1,4
2 1 4 3f f f f , f .
477. f 0,2 , 0 2 1f f 1 4f .
0 1x 0f x , :
) f 0,1 .
) f.
"1 1"
14
61
478. f , , 0 f f
0 2f . 0 0x 0f x :
) f 0, .
) f.
) 1 2, , , 1 2 1f f .
) 1 ,x , 122 2
f x f f
.
479. f 1,5 1 3f , 5 4f
1,6 . :
) 1 2, 1,5x x 1 2x x
1 2 0f x f x .
) 1,5 , 0f .
) 0 1,5x , 3
0 0 0 0f x f x f x x .
) 6y x fC
1,5 .
Fermat
480.
i) 2 4 1, 1,2 ) ( f x x x x ii) ( 1)) , 0,2( xf x x e x 481.
3 2
2
2 15 24 19 , [0,1]( )
18 1 , (1,2]
x x x xf x
x x x
f
482. 3 2 2f x x x x , , , x . , ,
f 1 1x 2 1x .
483. , , 2( )f x x x l x
x1 = 1 x2 = 3.
.
484. f 0, :
3 21
24
f x x xf x x 0 . f x0,
x0.
14
62
Fermat
485. :g , :
2 1xxg x e x x . 0 3g .
486. 1
2lnx
xx
0x , 2 .
487. f, , : , 1 2( ) 2,xe f x x x x (1) 2f .
fC (1,2)A
.
488. :f , :
2 3 4f x f f , x . :
) 3 4f f
) 3,4 ,
f ,M f x x .
489. 4 3 2
20074 3 2
x x xf x x
, , * , ,x .
f , 2 3 .
490. :f :
3 21 0f x x f x . f.
491. 2 2 2
f x x x x ,
.
492. , f , R,
2 2( () 1 2 )f x x xf x f .
[, ]
493. : 1,1f , 2 1f x x f x
1,1x . f .
494. : 0,1f 0 0f . f 0
14
63
495. : 2,4 f ( ) 0, ( ) 0f a f .
(,) ( ) 0f
M -
496.
( ).
) 4 26 7 2f x x x x ) 4f x x
) 2 1
2
x
x
ef x
e
) ( ) lnf x x x
497. :
) 2 3
2 3
2 , 0
4 , 0
x x xf x
x x x
)
2
3 2
4 6, x 0
6 4 6, 0
x xf x
x x x x
498. 4
1xf x e x , x .
) f .
) fC 0 0x .
) 4
1 3 2xe x x x .
499. 1ln xf x x e , 0x .
) f 0, .
) 2 3f x x 0x .
500. f :
1
3lim 4
1x
f x x
x
.
) fC 1, 1A f .
) A f , 7 4 0f x x x .
501. || < 1, f(x) = 4 3 2
3 112 3 2
x ax xx
R.
502. f (x) = x4 -x3 + 3
2 x2 -1
R.
14
64
503. 1
( ) ln , 0, 0f x a x x ax
.
, Cf
4
,0Ma
.
504. 3 22 1
2 53 2
f x x x x
, x .
, f 0
3
2x .
505. 4 3 2f x x x x x , 0 , x . fC
, 28
3 .
..
506. lnf x x x , 0x . ) f .
) 0 , 2
.
507. f, 0,5 2 0f .
3 0 2 5 0f f .
508. f . :
1 1f x f x f x f x , x .
f
509. f : R R ) (( 4) ( )f x f x f x
x .
2( ,) ( )xg x e f x x , .
510. g, , :
2 24 0g x x g x x x .
g .
14
65
511. 3 2 2 23 2 2 2 2 , x,f x x x x . f
, .
512. 23 6 xf x x e , ,x .
fC , .
513. f f x 1 f 3 x
x . f .
514. f f . f 1 0 ,
f .
515. f. A f
0x f .
516. f . (1) 0f ,
lim ( )x
f x
.
517. f ,
2
3lim 1
2x
f x x
x
4 6f .
) 2 6f .
) fC 2, 2A f .
) 3y x fC
0 2,4x .
) f , 2,4
f .
518. f 1,4 :
3 1 3
4
f ff x
2f x 1,4x . :
) 1 3f f
) 3 0f
14
66
) f 1,4
.
) 4 2f .
519. f , :
3 2 1
, 2 1
x
xf x x
.
) g(x) f (x) 3x .
) f y =3x
.
) f .
) lim 1x
f x f x
.
De L Hospital
0
0
520. :
) 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
521. :
) limxx
x
e ) lim
xx
nx
e )
(1 )lim
x
x
n e
x
522. :
) 2lim xx
e x x
) 2lim ln 2x
x x
) lim ln 1xx
e x
523. :
) lim ( )x
x nx
) 2lim ( )x
xx e
)
2lim ( 1)
x
nx 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
67
524. :
) 3
0lim( )x
x nx ) 2lim ( )x
xx e
)
1lim 1x
x nx
) 1
0limx
x
nx
e
) 1
0lim( )xx
xe
) 0
limx
x nx
)
1
0
1lim ( 1) ln 1
x
xx e x
x )
1
lim ( 1)xx
x e
) 3 2
0lim lnx
x x
0 01, ,
525. :
) 1
1lim 1
x
xx
,x>1 )
1
lim xx
x
) 1
lim ln xx
x
) 0
limx
xx
)
1
0lim 1 xx
x
) 1
lim x xx
e x
526. :
) 0
lim( 1)xx
e nx
) 1
lim
x
xe
x )
0lim(1 ) )
xx
e nx
De LHospital
527. 20
2 2 9lim
2
x
x
e x x x x
x
, , .
528. ln 1 , 0f x x x x x ) f .
) ln 1x x x 0x . ) f.
529. f, .
)
202 2 2 ( )
lim4h
f x h f x h f xf x
h
)
202 2 2
limh
f x h f x f x hf x
h
14
68
530. :f lim 2x
f x f x
.
limx
f x
, :
) lim 2x
f x
) lim 0x
f x
531. :f lim 1x
xf x f x
.
* limx
f x
, :
) lim 1x
f x
) lim 0x
f x
532. 2ln 2 ln 1x x x 0, .
533. ln , x 1,xf x x x e .
) f 1,+ .
) : x +
lnlim
x
x ,
x + x +lim , lim
xef x
x
) f (x) 1821
1,+ .
f
534.
:
) 2
3
2x
xf
x x
)
2x
5
xf
x
)
2 3 4x
2
x xf
x
) 2x 4 3 1 2f x x x ) ln
xx
fx
) x lnf x x
) 2
x1
x
x
ef
e
) x
x
xf
) x xf xe
535. 22 5 1
2
x xf x
x
y=2x-1.
536. 2
( ) 11
x
x
ef x x
e
. :
) y = x + 1 Cf -.
) y = x - 1 Cf + .
14
69
537. f 0, . y x , 0 Cf , :
) limx
f x
) limx
f x xf x
538. : f, g : RR g(x) = f(x) + x + x1
x x 0.
y = 2x + 3 Cf +,
Cg +.
539. : f, g : RR g(x) = f(xe-x) , x R.
y = 2x +1 Cf x0=0, Cg
+.
540. f ,
1 0f 2ln
1 1x
f xx
1x .
1y x fC .
541. 2
, x 2, ,2
x xf x
x
.. : y 2x 1
f , ,.
542. 2
xf x
x x
. , ,
f, 1x 4x .
543. 21 3x x
f xx
. , , ,
2x 2y fC .
544. 3 7y x f
.
)
limx
f x
x lim 3
xf x x
.
)
24
lim 23 3x
f x x
xf x x x
.
545. y = 4x - 3
f -, 2
2
( 1) ( ) 3 5 2limx
x f x x x
x x
.
14
70
546. , :f g , :
4f x g x x , x . 3 7y x fC .
) :
i)
limx
g x
x ii)
23 2x
lim3 1x
g x x
xf x x
) 2 3y x gC .
547. f ,g : 0,
2 lng x f x x x x 1x . f,g
1 2, ,
1 2 .
548. f,g :
( ) ( ) 3g x f x x .
f,g x=2 g
C
2y x , fC .
549. .
)2
( )2
xf x
x
)f(x) = 2x4-x2
550. :
) 2 2 2xf x e x x ) 2 xf x x e
) 1
xf x xe
551. ( ) 2nx
f x xx
) nx < x x > 0
) f
) f
Cf
) Cf : y = 3x.
) Cf , ,
y = 3x +1.
71
552. :
) 3f x x ) 41
f xx
,x>0
) 3f x x ) f x x x
553. :
) 21
, 0xf x x e xx
) 21 2 3
, 0,2
f x xx x x
) 21
2 , x 0,2
xf xx
554. :
) 1
ln xf x x ex
) f x x x x
) 1
11 xf x e
x
) 21 lnx
f xx
, 0x
) xf x e x x , 0x ) 2 33
x
x xf x
e
555. f :
) 2 33 1 3 5f x x x x 1
(0)2
f
) 1
2f x
x
x
72
558. f : . F
f , F(0)=0.
559. :f . F f R
, :
(0) 0F
2
2 ( ) f(x) e xxF x , x .
, f.
560. :f . F f R ,
:
(0) 1F
2
2
( 1)2 ( ) ( )
1
xf x F x
x
, x .
561. 5
1
( ) 2f x dx ,5
2
( ) 3f x dx 7
1
( ) 5f x dx , :
) 2
5
( )f x dx ) 7
5
( )f x dx ) 2
1
( )f x dx ) 7
2
( )f x dx