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To olzano
Bernard Bolzano(1781-
-
Weirstrass.
Otto Stolz.
eisatopon.blogspot.gr)
1
BOLZANO
f( ,
0x
0f(x ) 0
f(x) 0 .
f
A ,
x x. f
C
x x
.
- f
f .
- f
f ,
.
x x
0 x
y
α
f(β)
f(α)
0x βy΄
x΄
x
y
A(α,f(α))
α
Β(β,f(β))f(β)
f(α)
0x
β
y΄
x΄ 0
0 x
y
α
f(β)
f(α)
0x β
y΄
x΄
2
1. x2x x e x
0,2
.
x x
2x .
x
f(x) 2x 0,2
0
f(0) 2 0 e 2f 22 2 2 2
,
f(0)f 02
0x 0,
2
: 0x
0 0 0 0f(x ) 0 2x
0 x
x
2x .
2.
2 3x 1 x 1
0x 2 x 3
2,3 .
:
2 3x 1 x 1
0x 2 x 3
x 2
2x 1
x 3x 2
x 2 x 3
3x 1
x 3 0
2 3x 3 x 1 x 2 x 1 0
2 3f(x) x 3 x 1 x 2 x 1 2,3
2f(2) 2 1 0 , 3
f(3) 3 1 0
f 2 f 3 0 :
2 3f .
(
2 3
0(
2 3
0
.
2 3x 1 x 1
0x 2 x 3
3
3.
4 3f x x x 2x 1 3
g(x) 3x 5x 3
0
x 0,2 .
0
x 0,2
0 0 0 0
f x g x f x g x 0 .
4 3 3 4 3h x f x g x x x 2x 1 3x 5x 3 x 2x 3x 2 , x 0,2 .
0,2
h 0 2 0 4 3h 0 2 2 2 3 2 2 4 0 h 0 h 2 0 .
0
x 0,2
0 0 0 0 0
h x 0 f x g x 0 f x g x
4. f :
3 2 3 2f x f x f x x 2x 6x 1 x 2
3 , ,
f x 0 0,1 .
x 0 3 2f 0 2
f 0 f 0 (1)
2f 0
2 2
2 2 2
3 2
f 0 .
f 0 0 .
x 1 3 2 2f 1 (2)
2f 1
2 f 1 0 .
f 0 f 1 0 0,1
f x 0 0,1 .
5. 3 2f x x x x , 0 , 0
3 2 0 f x 0
0,1 .
f
3 2 3 2f x
2 2f x
2 2f x
4
2 2 .
f x 0
0,1 2 .
2g x .
g 0,1
g 0 g 1
g 0 g 1 0 Bolzano g x 0
f x 0 0,1 .
6. x2lnx 3e 1 0 0, .
xf x 2lnx 3e 1 , 0,1 .
x
x 0 x 0
lim f x lim 2lnx 3e 1 f .
f 1 2ln1 3e 1 3e 1 0 f
f Bolzano 0x
0x
0 0f x 0 2lnx 3e 1 0 .
7. f,g : 0, 0,
g(x) f(x) x , 0 f x 0 x 0
A1 1
x ,g x B2 2
x ,g x 1 2
x ,x 0,
1 2
x x AB / /x x .
AB / /x x 1 2g x g x .
1 2 2 1x x .
1 2x ,x 0,
1 1
1x 0, 1 1
g x g .
h x g x g .
0,
g 0 f 0 g .
g 0 g 1x 0,
1 1 1 1 1 1 2
h x 0 g x g .
8. 0,1 f,g 0,1
f g 0,1
5
0,1 2fog g f 2 1.
f xf x x 0,1 xf x , x 0,1 .
f 0,1
f 0,1 f 0 ,f 1 f 0 0 f 1 1 .
g 0,1
f 0,1 g 1 ,g 0 g 0 1 g 1 0 .
2fog 2
f g .
2h x f g x g f x 2x 1 , x 0,1 .
h 0,1
h 0 f g 0 g f 0 1 f 1 g 0 1 1 1 1 1 0
h 1 f g 1 g f 1 2 1 f 0 g 1 1 0 0 1 1 0
h 0 h 1 0 Bolzano
h 2fog
f x x x 0,1 f
0,1 f f x f x x f x f x x
x 0,1 f x x , x 0,1 .
9. f : ,
2f f f 0 , 0f .
2 2f
- f f , Bolzano
f .
- f .
f .
10. f,g : , , f g
0x ,
0 0 0
f x g x x ,
(2) x .
6
h x , x ,
h ,
h f g g g
g: , , g x x , g .
g 0 .
h f g f f
f : , , f x x ,
f . f 0 .
h h g f 0 .
h h 0 h 0 h 0 0
x 0
x .
h Bolzano 0x , : 0
h x 0 .
0 0 0 0 0x , : h x 0 f x g x x .
11. 0,1 f 0 f 1
0,1
1f f
4.
1
g4
0,1
0 x 1 0 x 1
1 1 30 x 1 x
4 4 4
3
0 x4
.
x 0 1
g 0 f 0 f4
.
1x
4
1 1 2g f f
4 4 4
.
2x
4
2 2 3g f f
4 4 4
.
3x
4
3 3g f f 1
4 4
.
1 2 3
g 0 g g g f 0 f 1 04 4 4
(1).
1 2 3
g 0 , g , g , g4 4 4
1 2 30, , ,
4 4 4
7
g 0 1 2 3
g , g , g4 4 4
1
g 0 g 04
1
0,4
1
4
g .
1
g4
.
12. 3x 3x 2 0 0,
3f x x 3x 2 , x 0,1 0,1
f 0 2 0 f 1 1 3 2 2 0 f 0 f 1 0 .
Bolzano 0x 0,1 0
f x 0 .
1 2x x 0,1
1 2x x
3 3
1 2x x
1 23x 3x
3 3
1 1 2 2x 3x x 3x 3 3
1 1 2 2x 3x 2 x 3x 2 1 2
f x f x
f 0,1 ,0
x
f x 0 .
13.
2 2 2
0x x 1 x 1
, , , 0 .
1,1 .
1 2,
2 2
2
1 2
1 1.
2 2 2
0x x 1 x 1
2 2 2 .
2 2 2f x
2f 1 2 , 2
f 0 0 2
f(1) 2 .
1,0 f 0 f 1 0
1 1
f .
0,1 f 0 f 1 0
2 2
f .
2 2 2 2 2 2f x
2 2 2 2 2 2 2f x
f x 0 1 2
8
1 2
2 2 2 2 2 2 2
2 2
1 2 2 2 2
2
1 2 2 2 2
2 2
2 22 2 2
2 1
2 2
1 2 1 2
2 2 2
1 1
.
14. 7 32x x 1 0
7 3f x 2x x 1, x .
1 2x ,x
1 2x x
3 3
1 2
7 7
1 2
x x
2x 2x
7 3 7 3
1 1 2 2 1 22x x 1 2x x 1 f x f x .
7
x xlim f x lim 2x
7
x xlim f x lim 2x
.
xlim f x
f xlim f x
,
f f
0x 0
f x 0 f
0
15. 3f x x x , 0 1
f x 0
f 0 , f 1 1 .
3 3
x x xlim f
f .
3 3
x x xlim f
f .
f f ,
f 0 f 1 0 f 1 f Bolzano
f x 0 .
f x 0 f
f
16. f : 0,2 f 0 0 , f 1 5 f 2 5 .
2f x 16 0,2 .
9
2g x f x 16
2g 0 f 0 16 16 0 , 2
g 1 f 1 16 25 16 9 0
2g 2 f 2 16 25 16 9 .
g 0 g 1 0 g 0,1
Bolzano, 2g x 0 f x 16
0,1 .
g 1,2 .
g 1 g 2 0 g Bolzano.
2g x f x 16 f x 4 f x 4 .
h x f x 4, x 1,2 h 1 f 1 4 5 4 1 0
h 2 f 2 4 5 4 9 0 h 1 h 2 0 .
h 1,2 Bolzano, 0x 1,2
0 0 0h x 0 f x 4 0 f x 4
2
0 0g x f x 16 16 16 0 g 1,2 .
17. f,g :
2f x x x g x x f x 0
g 0 g 1 0 .
g 0 g 1 0 0,1
0x 0,1 0
g x 0 2
0 0 0 0f x x x g x 0 .
0x 0,1 f x 0
f x 0 .
18. f : f x 0 x
f f f 0
f x 0 x f f f 0
f ,f ,f
f f f
, Bolzano
0x 0
f x 0
10
19. f,g
1f x f 2
g x g1 2
x ,x .
f,g
1 2f x g x
1 2x x .
1f x , 1 2
f x f x 2g x
g, 2 1g x g x .
1 2x x .
h x f x g x , 1 2
x x ,x .
1 1 1 2 1h x f x g x g x g x 0
2 2 2 2 1h x f x g x f x f x 0 1 2
h x h x 0 h
1 2x ,x Bolzano
h x 0 f x g x 1 2x ,x .
20. f f
x x
2.
fC AB / /x'x AB
2
0 0x ,f(x )
0x
2
.
AB2
0f x
2
.
0 0
B x ,f x2 2
.
AB / /x'x
0 0f x f x
2
0 0
0 02 2 2
.
0
2
0 x' x
y
y'
α β
Α Β
β α
2
0x
β α
11
0
x2
0 0f x f x
2
.
g x f x f x , x2 2
2
g2 2
g f f f f2 2 2 2 2 2
.
2
g2 2
.
- g2
g 0
2
0
x 0
x2
.
- g2
0x
2
0 0 0g x 0 f x f x
2
.
0
x2
0 0f x f x
2
.
21. f
2 2f x 3f x x x 5 x f
.
f
f f
0x 0
f x 0 .
0x x
2
0f (x )
0
03 f(x )
02 2
0 0 0 0x x 5 x x 5 0 .
f
.
22. 2f x x x 2 x .
f x 0 .
f .
2
x x 2 2x x 2 0 x .
fA .
2 2f x 0 x x 2 x 0 x x 2 x (1)
x 0
12
x 0 2
2 2 2 2x x 2 x x x 2 x x 2
f x 0
f x 0 x f .
f 1 1 1 2 1 1 0 f x 0 x .
23. f : f 0 3
2 2 2f x 6f x x 9 x x x .
2 2 2 2 2 2f x 6f x
2 2 2 2 2f x 6f x
2 22 2 2 2
f x 3 (1)
g x f x 3 .
2
x 9 0 x . .
x 0 g 0 f 0 3 g x 0 x .
2 2 2(1) g x x 9 g x x 9 2
g x x 9 g x 0
2 2 2g x x 9 f x 3 , x .
13
24.
2
2
2x x 3, 1 x 2f(x)
x 1, 2 x 3
.
Bolzano 1,3
f .
25.
4 3 2
x 8x x x 3 0 0,1 2x
x 0,2
5x 1 2 0, 6 4 2 2
x 8x 1,0
26.
10 20
x 1 x 10
x 2 x 2
2,2
xx e
0 0,2
27.0
x 0 : 0 0
2 2
0 0
21
1
.
28. 4 3 .
29. f : 0,
22lnx x f x ln x x x 0 . 2
f(x) e 1 lnx x
1,e .
30. f,g : 2,1 f x g x 3x
x 2,1 fC A 2,0 B 1,1
gC 2,1 .
31.2
g(x) . 2
2 .
32. f f f x x
f x x
33. f 2 2 2f f x f f x 3x
x f 2 2 .
f x 0 .
14
f 2
0x
0f x 0 .
34. 2f(x) x 5x 6 g x x 2,3
g 2 g 3 0 .
35. g: g ,
g x 0 x .
36. x1e 4 0
x 0,2 .
37.2x
lnx x e 0, .
38. 3 2f x x 2x 5
2g x x 3x 1 0 0
M x ,y 0x 1,2 .
39. f,g
f f
f .
40. f : 2f 2
f
0x 2
0 0f x 2x .
41. 2,3 0x 2,3
0
0 2
0 0
5 2xf x
x 5x 6
.
42. f,g , 0,1
10 f x 11 12 g x 13 x 0,1
2f .
43. 0,2 0 f x 1 x 0,2
2f .
44. 0
15
45. f : 0,1 f 0 f 1 0 f
0,1 .
46. 2f
f(x) 0 .
47. f : 0,1 0x 0,
2
0 0f .
48. f : f 1 f 5 1,3
f .
49.2 2
x
2
2 .
50. f : 0,2 0,1
2 2f .
51.x
e x 1821 0
52.
3
x 2x 4 0 1,2 4 3
x 2x 3x 8 0 1,3
3
x 0, 0,1 .
53. 2lnx 3 x 0
54. 5
x 24x 3 0
0,3 .
55.3 2
x 4x x 3 0
1,4 .
56. f : 1,1 f 1 2 , f 0 4 f 1 4 .
2f x 9 0 1,1 .
57. f 2 22f x 3f x x x 4
x f .
58. f 2 2xf x e x
f 0 1 .
16
59. f
2 2f x x 2f x x .
60. 2f x x 2x 4 x .
f x 0 .
f .
61. f : f(x) 0 x f(3) 5
2
x
f(4) 3 x 3x 1lim
x 5
.
62. f : f(3) 5
f(x) 0 3
2x
f(0)x 2x 5lim
x 4
.
63. 0,2
f 2 f 0 0 f f x 0 x 0,2 .
17
Bolzano
.Bolzano . Rolle
64. f : f( ) 0 ,
f( )f( ) 0 f( ) 0 , ,
f x x .
f x 0 f 0
f f 0 f 0 f f 0 .
f
Bolzano 1x 2x 1f x 0 2f x 0 .
f Rolle 1 2x ,x 1 2
f .
65. 2,3 2,3 .
19
f 3 f 23
f 2 6
0x 2,3 2
0 0f x x .
2,3 f 3 .
3
xh x f x
3
2,3 2h x f x x
8h 2 f 2
3
19 8
h 3 f 3 9 f 2 9 f 23 3
h 2 h 3 .
0x 2,3
2 2
0 0 0 0 0h x 0 f x x 0 f x x .
02,x 2,3 .
0
2
0 0 0 0 0 0 ,
0x 2,3 .
0 0
.
18
66. 3 26x 3x 5x 3 0
0,1 .
3 2f x 6x 3x 5x 3 , x 0,1 f
0,1 .
Bolzano.
f 0 3 0 f 1 5 0 f 0 f 1 0 f
0,1 f x 0 0,1 .
f 0,1 .
f 1 2 1 2 f
1 2 1 2 2f x 18x 6x 5 1 2f
Rolle 1 2
f 2
18 0 .
f 0,1
Bolzano
67. 2,2 f 2 2 f 2 6
0x 2,2 0 0
f x 4 x .
1 2, 2,2 1 2
f f 1 .
h x f x 4 x 2,2
h 2 f 2 4 2 2 4 2 4 0 h 2 f 2 4 2 6 4 2 4 0
h 2 h 2 0 0x 2,2
0 0 0 0 0
h x 0 f x 4 x 0 f x 4 x .
0 0
2,x , x ,2
1 0 2 0
0 0 0
1
0 0
f x f 2 4 x 2 2 xf
x 2 2 2 x
0 0 0
2
0 0 0
f 2 f x 6 4 x 2 xf
2 x 2 x 2 x (2)
19
0 0
1 2
0 0
2 x 2 xf
2 x 2 x
.
68. 1,e 1,e f 1 3
f e e 2 1 2, 1,e
1 2
1 2f f 1 .
e 2 3 e 1
1 e 1 e
.
h(x) f(x) x e 3
1,e
h 1 f 1 1 e 3 3 1 e 3 1 e 0 ,
h e f e e e 3 e 2 3 e 1 0
h 1 h e 0
0x 1,e
0h x 0
0 0
f x x e 3 0 0 0f x x e 3.
01,x
0x ,e
1 0 2 0
0 0 0
1
0 0 0
f x f 1 x e 3 3 e xf
x 1 x 1 x 1
0 0 0
2
0 0 0
f e f x e 2 x e 3 x 1f
e x e x e x,
0 0
1 2
0 0
e x x 1f
x 1 e x
.
69. f , f f 0
1 2, ,
1 2 1 2
f f 0 .
f f 0 f , Bolzano 0
x ,
0f x 0 .
f 0,x 0
x , 1 0
,x 2 0
x ,
0
1
0 0
f x f ff
x x
0
2
0 0
f f x ff
x x
x
y
y΄x΄ 0
3
e
e+2
0x
A
BΓ (1,e+2)
Δ (e,3)
1
20
1 2
0 0 0 0
f f f ff f 0
x x x x
70. f , , f f .
, f f f f
.
1 2, ,
1 2
2
1 2
f ff f
.
f f f f f ff f
f ff f 0 (1)
f fg x f x f x , x , g ,
f fg f f f f
f fg f f f f
2
g g f f 0 ,
Bolzano ,
f fg 0 f f 0
f f f f.
f , ,
, , 1
,
2
,
1
f ff
2
f ff .
f ff f
1
f ff f
f ff
2
f f f ff ff f
f
21
2
f f f ff
1 2
f ff f
f f
2
f f
71. f 2,2
2
2 xf x 1 f x 1 x 1 e 2 x
2,2 f 0 .
x 1 f 2 f 0 2 x 1 f 0 f 2 2 .
f 1
x 2,0 2
x 0,2 1
f x 1
2
f x 1 f 1 2x ,x
1 2f x f x 0 Bolzano
1 2
x ,x 2,2 f 0 .
Bolzano
72. f,g
f x g x x f 0 g 0
f x g x cx , c , x .
f x 0 1 2
g x 0
1 2, .
f x g x f x g x c g x cx 1
f x g x cx c
1
f 0 g 0 c 0 f x g x cx .
1 1
g f 0
1 1c c ,
2 2g f
0
2 2c c , 2
1 2 1 2g g c 0 .
1 2
g g 0 g 1
2.
1 2
g g 0 g x 0 1 2, .
73. f 0,
2 xxf x f x x e x 0 f 1 e 1
x
f xg x e
x 0, .
22
xf x x e 1 , x 0 .
f x 2 0,1 .
2 x
x x
2 2
xf x f x x eg x e e 0 g x c
x x
x xf x
f 1 e 1 c 1 e 1 f x x e 1x
g x f x 2 .
x
x 0 x 0
lim f x 2 lim x e 1 2 2 0
g 0 g 1 f 1 2 e 1 0
g g 1 0 g x 0 f x 2
,1 0,1 g 1,
20,1
1 2.
g 1 2,
1 2,
x xg x f x e 1 xe
1 2g g 0 Rolle
1 2,
g 0 e 1 e 0 e 1 e 0 .
g x 0 f x 2 0,1 .
Bolzano
74. f : 0, 2x
2 f x 4e
x 0 , f 0 0 f 0 1 .
2 2xx 1 f x e x 0 .
0x 0,1 0 0
f x lnx 0
2xg x f x e , x 0 .
g 0, 2xg x f x 2e
2xg x f x 4e .
2x2 f x 4e 2x
f x 4e 0 g x 0 g
0, .
g
x 0 g x g 0 0
g 0, .
g
2x 2xx 0 g x g 0 0 f x e 0 f x e
.
2h x f x x 1, x 0 .
h 0, h x f x 2x h x f x 2 .
23
2x2 f x 4e f x 2 0 h x 0 h
0, h
x 0 h x h 0 0
h
0, .
h
2 2x 0 h x h 0 0 f x x 1 0 f x x 1
.
x 0 x 0
lim
2 2xx 1 f x e x 0 2
2 f 1 e f 1 0
0
f 1 0
Bolzano 0
0 0 0
0,1 1
.
h x 0
f x 2x 0 f x 2x 0 x 0,1
0x f x lnx 0
0,1 .
75. f f x 0 x ,
f f f 1
f , .
2f x x x , .
f x 0 f
f
0x 0f x 0
f
0 0x
f x 0 x ,
f
f f x 0 x f
.
2g x f x .
g g x f x .
f
g
.
g
2
0
Bolzano
24
2 2f x
76. 4 3 2x x 20x 28 0
1,2 .
4 3 2f x x x 20x 28
1,2 .
f 1 6 0 f 2 76 0 f 1 f 2 0 .
Bolzano 0x 1,2 0
f x 00
x
3 2f x 4x 3x 40x 2
x 4x 3x 40 0 x 1,2 .
f 1,2 0x
.
77. f f x f x
e 4x 4e 0
x .
f.
f
2
f x f x f x f x f x
f x
4e 4x 4e 0 e 4x 0 e 4xe 4
e
2 2f x f x f x2 2 2
e 4xe 4x 4x 4 e 2x 4x 4 (1)
x 0
2f 0e 4 2f 0 ln4 f 0 ln2
24x 4 0 x
f xe 2x 0 .
f xe 2x
f 0e 2 0 2 0
f xe 2x 0
f x f x2 2e 2x 4x 4 e 4x 4 2x .
2 2 2 2 24x 4 4x 4x 4 2x 4x 4 2x 4x 4
24x 4 2x 0 x 2
f x ln 4x 4 2x .
2
2f x 0 f
4x 4.
25
78. f : 0,
f 0 1 , f 0 2 2
f x f x f x x 0
2xf x e x 0 .
f xg x
f x
2
f x f x f x f xg x 0 g 0,
f x .
x 0
f x f 0 f xg x g 0 2
f x f 0 f x.
f x 0 f f 0 1 0 f x 0 .
2x 2x 2xf x 2f x f x 2f x 0 e f x 2e f x 0 e f x 0
2xh x e f x
h x 0 h 0,
x 0 2x 2x 2xh x h 0 e f x f 0 e f x 1 f x e
79. f 1,3 f 1 3 f x 4
x 1,3 f x 0 1,3 .
g x f x 4x, x 1,3 g x f x 4 0 g 1,3 .
g
1 3 g 1 g 3 f 1 4 f 3 12 f 3 5 .
f 1 f 3 0 f 1,3 f x 0
1,3 f x 4 0 f 1,3
80. f : 0,
x2x f x e x 0 f 0 2 .
2 xx 2 f x e 1 x 0 .
f.
2f x 3x 1,2 .
2g x f x x g x f x 2x 0
g 0,
x 0 2 2g x g 0 f x x 2 f x x 2 .
xh x f x e ,x 0 x
h x f x e 0
h 0,
x 0 x xh x h 0 f x e 1 f x e 1.
2 xx 2 f x e 1 1
26
f x 2x 0 0, f
2 x
x xlim x 2 lim e 1
xlim f x
xf A f 0 , lim f x 2,
2x f x 3x
1 f 1 3 0 2f 1 1 2 f 1 3 , 2 f 2 12 0 2
f 2 e 1 12
Bolzano
Bolzano
81. f : , f x 0 x ,
f , .
f
.
1 2x ,x 1 2x x
1 2f x 0 f x .
f 1 2x ,x
f 0x 1 2x ,x
Fermat 0f x 0
f 1x 1 1f x f x f x f x 0
1 2x x ,x 1 2x x x 1x x 0 ,
1
1
f x f x0
x x
1
1
x x1
f x f xlim 0
x x
1f x 0
f 2x 2 2f x f x f x f x 0
1 2x x ,x 1 2x x x 2x x 0 2
2
f x f x0
x x
1
2
x x2
f x f xlim 0
x x
2f x 0
1 2x ,x 1 2x x
1 2f x 0 f x f
f
Bolzano
27
82. f 1,4
f 3 f 4 f 1 f 2 f
f 1,4 f 1 f 3 f 2 f 4 f
f 1,4
f 1 ,f 4
1 2
x ,x 1,4 Fermat
1f x 0 2
f x 0 f f
f 1 f 3 f 2 f 4 f
1,4
Fermat.
f 1,2 ,
2,3 , 3,4 1 , 2
3
1
f 2 f 1f
2 1
,
2
f 3 f 2f
3 2
3
f 4 f 3f
4 3
.
f 1 f 3 f 2 f 4 1f , 2
f 3f .
1 2f , 2 3
f f 1,4
Bolzano, f x 0
1 2
2 3 f
83. xf x e , g x x h x lnx , x 0
x , 0 f, h
f x g x h x x 0 .
AB d .
0
0,1 d
xx e x , x
x e 1 0 x 0
x 0 x 0 1 0 f x g x .
x x lnx ,
1 x 1
x 1 0 x 1x x
x 1 1 1 0 x 1 0 x lnx .
28
xe x lnx .
A ,e B ,ln
2 22
d e ln e ln e ln e ln
1d e ,
2
1d e 0 d 0, .
d 1 e 1 0
0 0
1lim d lim e
0,1 : d 0 o lzano ,1 0,1
0 0
,1 0,1 : d 0 d
0
d
0 0 0x d x d 0 d 0,
d
0 0 0x d x d 0 d , d
0.
84. f t
t t 0 f t
82
t 1.
f t .
t
;
t
t
ln11 2,4 )
8
f t 2t 1
f t 8ln t 1 2t f t 8ln t 1 2t c, t 0,c
t 0
f 0 0 .
f 0 c c 0 t f t 8ln t 1 2t, .
8 6 2tf t 2 0 t 3
t 1 t 1.
t 0,3 f t 0 f 0,3 t 3
f t 0 f 3,
t 8
2 3f 8 8ln9 16 8ln3 16 16ln3 16 16 ln3 1 16 ln3 lne 16ln
e .
3 3
1 ln 0e e
f 8 0 .
t 8
29
f 0 8ln11 20 8 2,4 20 0,8 ~
.
f 8 f 1 0 f
Bolzano 1
t 8,10 1f t 0 .
t 10
85.
3 2 3 2f x f x f x x 2x 6x 1 x ,
23 .
f
f
f x 0
0,1 .
f 0
x .
f Fermat 0
f x 0 .
3 2 3 2f x f x f x x 2x 6x 1
3 2 3 2
f x f x f x x 2x 6x 1
2 23f x f x 2 f x f x f x 3x 4x 6
2 2f x 3f x 2 f x 3x 4x 6 (2).
0
x x
2 2 2
0 0 0 0 0 0 0f x 3f x 2 f x 3x 4x 6 0 3x 4x 6
56 0 f
23x 4x 6 56 0 2
3x 4x 6 0
x 23f x 2 f x
2 24 12 4 3 0 2
3f x 2 f x 0 x .
f x 0 x f
.
x 0
3 2f 0 f 0 f 0 1 2
f 0 f 0 f 0 1 (3).
2f 0 f 0 2 2
4 3
2
23 0
3 2
3 0 0
2f 0 f 0 0 f 0 0 .
x 1
3 2f 1 f 1 f 1 1 2 6 1 2
f 1 f 1 f 1 4 .
2f 1 f 1 2
4 0
2f 1 f 1 0 f 1 0 .
30
f 0 f 1 0 f 0,1 Bolzano
0
x 0,1 0f x 0 .
f 0
x
f x 0 0,1 .
86. f 1,5 f 1 3 , f 5 4
1,6
1 2
x ,x 1,5 1 2
x x
1 2
f x f x 0 .
1,5 f 0 .
0
x 1,5 3
0 0 0 0f x f x f x x .
y x 6f
C 1,5 .
1 2
x ,x 1,5 1 2
x x
1f x 1 f , 2
f x 6 f
Fermat 1 2
f x f x 0
1 2
f x f x 0 f 1 2x ,x
1 2x ,x Rolle 1 2
x ,x 1,5 f 0 .
4h x f x f x f x x , 1 2
x x ,x 1,5
4
1 1 1 1 1 1h x f x f x f x x 1 x 0 4 4
2 2 2 2h x f x x 6 x 0
h 0 1 2
x x ,x 1,5 : 0h x 0
g x f x x 6 f x x 6 , x 1,5
g 1 f 1 1 6 3 1 6 0 , g 5 f 5 5 6 4 5 6 0 g
1,5 : g 0 f 6 .
Bolzano
87. f : f x2
x x 1 f x xe 0
x
4 2 3f x 90x 150x 60x 180x ,
3 2 3 2f x 360x 300x 180x 180 60 6x 3x 5x 3
fC 0,1 f x 0
31
f 0 180 0, f 1 300 0 f
Bolzano 0,1 f 0 .
3 2f x 60 18x 6x 5 0 0 f .
x f x f 0 f 0, x
f x f 0 f ,1f
C
0,1 ,f .
88. f
f .
f , , f
y x
f f
y x
g x f x x , x g x f x g f 0 .
f f g
.
x g x g 0 g
, x g x g 0 g
, g x , g x g
x .
g f 0 g x g 0 f x x 0
f x x x Cf y x
f g f 0 .
1
x,
1 1
g g xg g x g x g
x.
1
g g 0 k 0 1
g k .
1
g x g k x g
xlim k x g
1
x xlim g x lim g x g
, g 0 g
Bolzano , 1
x , 1g x 0 .
,x 2
,x
2 2
g x gg g x g x g
x
32
g
2 2g g 0 m 0
2
g m 2
g x g m x g
x xlim m x g lim mx
2x xlim g x lim g x g
, g 0 g
Bolzano , 2
x , 2g x 0
g x 0 f x x .
89. f ,
, 0 f f
, , , 0f f
f x 0 , .
, , 0f 0f .
f.
f f 0 2f 0
.
f , f f 0 Bolzano
0x , , 0
f x 0 .
f Rolle
0,x 0
x , 1 0
x ,x 2 0
x x ,
1 2
f x 0 f x (1).
f , 3
x ,
3
f f ff x
.
f f 0 f , f
f 0 f 0 3
f x 0 .
f 0,x
4 0x ,x
0
4
0 0
f x f ff x 0
x x
.
f 0x ,
5 0x x ,
0
5
0 0
f f x ff x 0
x x
.
f , 6
x ,
6
f f ff x 0
.
33
f 3 4x ,x
2 3 4x ,x
4 3
2
4 3
f x f xf 0
x x.
f 5 6x ,x
1 5 6x ,x
6 5
1
6 5
f x f xf 0
x x.
1 2
f f 0 f 1 2,
Bolzano 1 2, , f 0 f
.Bolzano
90. : y x , 1
0e
f x lnx
1,e e, .
lnx x
g x lnx x , x 0 .
g 0 , g e 1 e 0 g 1 g e 0 .
Bolzano 1 1
x 1,e : g x 0 .
x x x
lnxlim g x lim lnx x lim x
x
x x
1
lnx xlim lim 0x 1
e g 0 .
g e g 0 Bolzano 2 2
x e : g x 0 .
g 3 2 1
x x x Rolle
1
g x 0x
1 10 x
x
g x 0
.Bolzano
91. 2f x x 2lnx, x 0 .
) f x 1 x 0 .
) f.
34
)
ln x, x 0
f xg x
k, x 0
.
i k g
ii 1
k2
g
0,e .
) f 0,
22 x 12
f x 2xx x
.
2x 0 x 0
22 x 1
f x 0 0 x 1 x 1x
.
x 0,1 f x 0 f
0,1 .
x 1, f x 0 f 1, .
f 2f 1 1 2ln1 1 f x 1 x 0 .
)
2
x o x o
lim f x lim x 2lnx
x o
lim lnx x 0 ,
y y f
C .
x x
f x lnxlim lim x 2
x x
x DLH x x
1
lnx 1xlim lim lim 0x 1 x
fC
2 2
2x x x
lnxlim f x lim x 2lnx lim x 1 2
x.
2 2x DLH x x
1
lnx 1xlim lim lim 02xx 2x
xlim f x
fC
) i. g 0, g
x o
lim g x g 0 k
2 2 2DLHx o x o x o x o x o
1 1
lnx 1 1x xlim g x lim lim lim lim2 2x 2lnx 2x 2 2x 2
2xx x
1
k2
.
ii.
2 2
lne 1 1g e 0
f e e 2lne e 2
1g 0 k 0
2 g 0 g e 0
g 0,e Bolzano
g x 0 0,e .
x 0 1
f - +
f
] [
35
92.
f 3 0 f 4 f 5 f 4 0f
93. 1,2 .
3x 2 3lnx 4 3 2
x x 30 20x
94. x x x ( x) ( x) ( x) , , ,
95. -
f f x
g f x
f x
ee
e x
i. f x g x x x .
ii. f x 0 , .
,f , ,
96. f : 0,3 f 3 f 1 f 2 f 0 .
1 2 3 1
f , 2f
3f .
f .
97. , 1,2 f 1 4 f 2 1.
0x 1,2
0 0f x 3x 2 .
1 2 1 2
f .
98. f f f
f
2x y 1 0 .
f
f x .
1 2
1 2
1 2
1 11
f
.
36
99. f 0,1 . f 0 2
f 1 4
y 3f
C 0x 0,1 .
1x 0,1 1
1 2 3 4f f f f
5 5 5 5f x
4
.
2x 0,1
fC 2 2
y 2x 1821 .
100.α f x x x0
x
,6 4
.
0x ,
4
0
f x f4 4
i) 3
f2
ii)
0
2f x
4 3 4 4
101.v
x lnx 1 , v * 0,1 .
102. 2 2
15x f x 5x
x 3, f 5 0 f
f x 0 3, .
f 3, .
103. f : 1,e 0,1 f x 0 x 1,e
f x x 1 lnx 1,e .
104. 3f x x 2x 1 ,
3 0,1 .
105. f 1,2 f 1 1
f x 1 x 1,2 f(x) 0
1,2 .
106. f 3f x 2f x 3x 0
x .
f
f 1 .
37
1,1
2 f 3 4
107. f 0,8 f 2 f 0 f 8 f 6
f
f 0,8 f 8 f 2 f 6 f 0 f
108. f :
x xlim f x lim f x f x 0 x .
0
x 0
f x 0 .
f
109. f 0,2 f 0 f 2 1 f 1 4
0 f x 0
f 0,1 .
f f.
110. f
3 3 2f x f x 8x 12x 8x 2 x .
f -1.
f x 0 0,1 .
g : 2f g x 3x f x 2 x
0x g .
111. f 0,2 5,5
f 0 1, f 2 4 .
1 2x ,x 0,2 1 2f x f x 0
f .
1 2 1 2f .
112. f f
0,6 0x 3 f x 0 .
i. f 2,3 .
ii. f 6 3f 2 0
iii. 3f 3 f 6
f 6 0 0,3
f e f 3 ln f ln f 3 e
38
113. f :
2f x 2x f 2 f 2 x
f 2 f 2 4
0x 2,2 0
f x f 2 3 .
1 2, 2,2 1 2
f f 2 .
2,2 f 1 .
f x 1
114. 0 ln 2 0 .
f : 0,
1f x 1 lnx 1
x.
i. f
ii.
21
f x 0 .
iii. f f 0
115. f
f 0 0 , f 1 0 , f 2 0 f x 0 x 0,2
f x 0 0,2 .
f 0,2 .
f x f 0 x f 0 .
116. f f f
f 0x 0
f 2 f 2 0 f 2 f 2 0 f 1 f 0
0x 2,2 2
0 0f x x 4 .
1 2 3 4 1 2 3 4
f .
117. xf x , x 1 , .
f x 1 x 1 .
,
i. f
ii. f 1,0
0, .
iii. f
0
1,2 .
39
118. f 2 0,1 1,10 ,
f 0 2 f 1 7
1 2
x ,x 0,1 1 2
f x f x 0 .
1
0,1 1
f 0 .
2
0,1 2 2
f f 0 .
f y 15x 6
0
x 0,1 .
119. f 2 , 1 f 1 ,
f 1 2 1
f2
,
f , 1
x 2y 2 0 f.
1 1
f x x 12 2
x , 1 .
, 1 1
f2
.
f : y 3x 4 2
, 1 .
120. xf x e 1 2
h x x x 1.
f O 0,0
h.
1,0 e 2 1 0 .
g x f x h x
i. 2g x 1 x .
ii. g x 1821
40
Bolzano
121. f f x 4f 2x 1
x , f 1 0 1
f 12
1 1
1 0f x dx 2 f x dx 2
0,1 f f 2 1 e 2 .
1
f x 4f 2x 1 f x 2f 2x 1 c x 1
1 1
f 1 2f 1 c c 0 2
f x 2f 2x 1 f x f 2x 1 c .
x 1 2 2
f 1 f 1 c c 1 f x f 2x 1 1 x .
1 1 1
0 0 0f x dx f 2x 1 dx 1dx 1.
u 1
2x 1 u x2
1
dx du2
x 0 u 1 x 1
u 1
1 1
0 1
1f x dx f u du 1
2
1 1
1 0f x dx 2 f x dx 2
g x f x f 2x 1 e 2x 1 e 2x , x 0,1 .
g 0 1 e 0 , g 1 3 e 0 0,1
0,1 g 0 f f 2 1 e 2 .
122. f 1,3 3
1f x dx 0
0x 1,3 0
f x 0 .
f x 0 1,3 .
f 1,3 f 1 f 3 0 ,
1 2
x ,x 1,3 1 2
f x f x 0 .
f 1,3
- 0f x 0 f x 0 x 1,3
3
1f x dx 0
- 0f x 0 f x 0 x 1,3
3
1f x dx 0
f 1,3 1 2, 1,3
1 2f f 0
f Bolzano 1 2, 1,3 f 0 .
f 1,3 f x 0 x 1,3
f 1,3 1 1 f 1 f 3 f 0 Rolle
1
1, 2
,3 1
f 0 2
f 0
41
1 2
f f 0 f 1 1 1 2
f
1,3 1 2
x ,x 1,3 1 2
f x f x 0 .
123. f
x
2
0f x f t dt x x x
x x
2
0 0g x f t dt f t dt x 2
x x
2
0 0f t dt f t dt x x .
0,1
3
0 0f t dt f t dt 1 2 .
f x f x 1
x
2
0f x f t dt x x 2
f x
0f t dt ,
x
0f t dt
x x
2
0 0g x f t dt f t dt x 2
x x
0 0g x f x f t dt f x f t dt 2x 3
2 x x x
2
0f x f t dt x x 4
3 2 4 2 2g x x x x x 2x 2x 2x 0
g x c g 0 2 g x 2 , x
x x
2
0 0g x 2 f t dt f t dt x 2 2
x x
2
0 0f t dt f t dt x , x
x x
3
0 0x f t dt f t dt 2x 1 0,1
0 1 0
1 1
2
0 01 f t dt f t dt 1 1 1 2 0 0 1 0 Bolzano
0,1
3
0 00 f t dt f t dt 1 2 .
124. f 0,2 f x 2 x 0,2
0,2
02 3 f t dt .
x
0g x f t dt 3x 2 , x 0,2 .
42
f 0,2 3x 2 g
0,2
0
0g 0 f t dt 3 0 2 2 0
2 2
0 0g 2 f t dt 6 2 f t dt 4 .
f x 2 f x 2 0 2 2
0 0f x 2 dx 0 f x dx 4 0 g 2 0 .
g 0 g 2 0 Bolzano 0,2
0g 0 2 3 f t dt .
x
0g x f t dt 3x 2 f x 3 .
f x 2 f x 2 0 f x 3 1 0 g x 0 g
0,2
x
0g x 0 2 3x f t dt 0,2 .
125. f,g
x
0f x x 1 g t dt x
3g x
4 x 0,4 g
0,4 .
0,4
0g t dt 1 .
0,
0g g t dt 0 .
f 0,4 f 0 1 0 , 4
0f 4 3 g t dt (1).
3 3
g x g x 04 4
g
0x 0,4 0
3g x
4
4 4 4
0 0 0
3 3g x dx 0 g x dx dx 0
4 4
4
0g t dt 3 0 f 4 0 .
f 0 f 4 0 Bolzano 0,4
0 0f 0 1 g t dt 0 g t dt 1 (2).
f 0,4 3 1
f x 1 g x 1 04 4
f
0,4
f 0, 0,
0,
f f 0f f f 0 1 g 1 g 1
43
2
0g 1 g g t dt
0g g t dt 0 .
126. f,g :
x 1 2x 3
x 2
1 1f t dt g t dt e 1
x
2 2 x 22 2x x f x x 3x g x e
0,1 .
x 1 2x 3 x 1 2x 3
x 2 x 2
1 1 1 1f t dt g t dt e 1 f t dt g t dt e 1 0
.
x 1 2x 3
x 2
1 1h x f t dt g t dt e 1
, x .
1 1
1 1h 2 f t dt g t dt 1 1 0 .
h x h 2 , x h
0x 2 .
f,g x 1 , 2x 3 , x 2
e 1
h
x 2h x f x 1 2g 2x 3 e
Fermat
h 2 0 f 1 2g 1 1 0 f 1 2g 1 1 .
2 2 x 2x 2 2x x f x x 3x g x e , x 0,1 .
2 1
1 1
e e .
0,1 Bolzano
2 2 x 2 0,1 .
127. f , f f x
dx 0
1 2, , f
2
f x dx 0 .
f f x dx 0 .
f 0 1
f x dx 0 2 0
x , 0f x 0
f x 0 x ,
f x dx 0 f
00, x
0f f x 0 Bolzano
1 0 1, x , : f 0 .
E
t
h t f x dx h 0
h f x dx 0 2 .
44
h , h t f t h f 0 1
x x
h x h h xlim 0 lim 0
x x
1x
1
1
h x0
x
1h x 0 .
h 1x ,
1h x h 0
Bolzano 2 1
x , ,
2
2h 0 f x dx 0 .
f 0
f x dx 0 .
128. f , f f 0
f x dx 0
,
f x dx 0 .
x
g x f t dt , x , f
x
g x f t dt g x f x .
g 0 ,
g f t dt 0 g f , g f
f f 0 g g 0 g , g
- g 0 g 0
x x
g x g g xg 0 lim 0 lim 0
x x
1
x 1
x
1
1
1
g x0 g x 0
x.
-
x x
g x g g xg 0 lim 0 lim 0
x x
2x
2
x
2
2
2
g x0 g x 0
x.
g 1 2x , x ,
1 2g x g x 0
Bolzano 1 2
x , x , ,
g 0 f t dt 0
129. f 0,2 1 2
0 1f x dx f x dx .
f -1
0,2 1
1f f t dt .
10,2
1
0 0
3f t dt f t dt
2
45
x 1 x 1 x
x 1 1g x f t dt f t dt f t dt f
g g x f x 1 f x , x 0,1 .
1
0g 0 f t dt ,
2
1g 1 f t dt g 0 g 1
g Rolle 0,1
0
x 0,1 0 0 0
g x 0 f x 1 f x f 1 1.
1
x
1f x f xt dt 0,2
x 0 : 1
x
1xf x x f xt dt u xt du xdt
1 1 1
x x xxf x f u du xf x f u du 0 x f u du 0
1
xh x x f u du , x 0,1 0,1
h 0 0 , h 1 0 h 0 h 1 Rolle
0,1 h 0 .
x 1
0 0
3x f t dt f t dt
2, x 0,2 f
1
0
30 f t dt
2,
2 1 1 2 1 1
0 0 0 1 0 0
3 12 f t dt 3 f t dt f t dt f t dt f t dt f t dt
2 2
2
1
0
30 2 f t dt 0
4
- 0 2 0 1
0 1
2 x 0
- 0 2 0 Bolzano 1
0,2 1
0
1 1
1 11 0
0,2 : 0 f t dt 3 f t dt
130. f 1,
x
e
f tdt
tf x e , x > 1.
1, .
2xf x f x ,x 1 .
1
f xln x
.
0 0M x , f x 2
0x e,e ,
Cf
15
2
t 1 1
x
u x 1
46
x 1
xxlim f t dt 0 .
f f t
t
x
e
f tg x dt
t
g x
f x e
1,
1
x
e
f tdt x
t
e
f tf x e lnf x dt
t
2
f x f xxf x f x
f x x, x 1.
2
2
f x 1 1 1xf x f x lnx lnx c
x f x f xf x 2 .
1 x e 0f e e 1 2 x e c 0 .
1
f xlnx
, x 1.
0 0M x ,f x
0 0 0y f x f x x x
0 0 0 0
y f x x f x x f x .
0 0 0
0
f x x f xA ,0
f x
0 0 0B 0, f x x f x .
0 0 0
OAB 0 0 0
0
f x x f x1E f x x f x
2 f x
0
0
2f x 0
20 0 0
0 0 0 0x e
0
f x x f x15 115f x f x x f x
2 2 f x
2
h x f x xf x 15f x 2e,e .
f e 1, 1
f ee
, 2 1f e
2 2
2
1f e
4e.
15 4e 15
h e 4 0e e
2
2
2 2
9 15 9e 60h e 0
16 4e 16e
2 23e 8 9e 64 9e 60 0 , Bolzano 2
0x e,e
0h x 0 .
x
1F x f t dt f t , F x f x .
F x,x 1 x,x 1
x 1 x x 1
1 1 x
F x 1 F x 1F f f t dt f t dt f t dt
x 1 x ln 3 .
x x 1 lnx ln ln x 1
47
3x 1
x
1 1 1 1 1f t dt
lnx ln ln x 1 lnx ln x 1.
x
1lim 0,
lnx
x
1lim 0
ln x 1
x 1
xxlim f t dt 0 .
131. f : x
f x x
x
0
tf x x 3 dt
f t t.
f
f xf x
f x x,
x .
2g x f x 2xf x , x
2f x x x 9, x .
x 1 x 2
x x 1
f t dt f t dt x .
f f t t
t
f t t
x
0
tdt
f t t
x
0
tf x x 3 dt
f t t
x
0
f x x x f xt xf x x 3 dt 1
f t t f x x f x x f x x
f , 2f x 2xf x
g
2
g x f x 2xf x 2f x f x 2f x 2xf x
f x f xg x 2f x 2f x 2x
f x x f x x
22f x 2f x f x x 2xf x
g xf x x
2 22f x 2f x 2xf x 2xf x
g x 0 g x c, cf x x
48
0
0
tf 0 0 3 dt 3
f t t 2
g 0 f 0 2 0 f 0 9
g 0 c c 9 g x 9 x .
2 2 2 2g x 9 f x 2xf x 9 f x 2xf x x x 9
2 2f x x x 9 (1)
f x x h x f x x 0 x h
h 0 f 0 0 3 0 ,
h x f x x 0 x
2 2f x x x 9 f x x x 9, x .
f
2
2
2
x 9 2f x x x 9 1 1
2 x 9
x
2
2
2 2
x 9 x
x 9 x 9.
2 2 2 2 2x 9 x 0 x 9 x x x x 9 x 0 x ,
f x 0 f .
x t x 1 f
f x f t f x 1
x 1 x 1 x 1
x x xf x dt f t dt f x 1 dt
x 1 x 1 x 1
x x xf x dt f t dt f x 1 dt
x 1
xf x x 1 x f t dt f x 1 x 1 x
x 1
xf x f t dt f x 1 (1)
x 1 t x 2 f
f x 1 f t f x 2
x 2 x 2 x 2
x 1 x 1 x 1f x 1 dt f t dt f x 2 dt
x 2 x 2 x 2
x 1 x 1 x 1f x 1 dt f t dt f x 2 dt
x 2
x 1f x 1 x 2 x 1 f t dt f x 2 x 2 x 1
x 2
x 1f x 1 f t dt f x 2 (2)
x 1 x 2
x x 1f x f t dt f x 1 f t dt f x 2 ,
x 1 x 2
x x 1f t dt f t dt
132. f : (0, ) x 0
f x 0 ,
2 2
x x 1
1
x xf t dt
e
x
1
ln t tln x x dt e f t
f t
xf x e nx x- , x 0
2
x 0
1lim f x f x
f x
lnx x- x 0
x
F x f t dt , x 0 0
49
F x F x F x x 0 .
0
, F F F
2 2
x x 1
1
x xh x f x dt , x 0,
e
2 22 2x x 1 x x 1
1 1x 0 h x 0 h x h 1
x x x xf(t)dt f dt
e e
h x 1 .
h 0,
2 1 2xh x f x x 1 2x 1
e
x 1 h Fermat
1 1
h 1 0 f 1 0 f 1e e
.
f 0, f(x) 0 f 0, .
1
f 1 0e
f x 0 x 0, .
x
1
lnt tlnx x dt e f t
f t
x x
1 1
lnt t lnt tlnx x dt e f x lnx x dt e f x
f t f t
lnx x x 0, f x 0
x
1
lnt tdt e 0
f t
x
1
lnx xf x
lnt tdt e
f t
.
lnt t
f t
0,
x
1
lnt tdt
f t
lnx x f
0,
x x
1 1
lnt t lnx x lnt tlnx x dt e f x dt e
f t f x f t
(1)
lnx xh x , x 0
f x
x
1h x h t dt e .
h 0, x
1h t dt
0, h h x h x
x xh x h x 0 e h x e h x 0
50
x x xe h x 0 e h x c, c h x ce
1
1h x h t dt e e ce e c 1 ,
x xlnx xh x e e
f x
x
f x e lnx x , x 0 .
1
f x ,
x
xx 0 x 0 x 0 x 0
1 1 elim
f x lnx xe lnx x
x
x 0
lim e 1
x 0
lim lnx x
0
02
2 2 DLHx 0 2
1 1 1lim f x
f x
lim lim 02
F 0, F (x) f(x), x 0
x x x1 1F (x) f (x) e lnx x e 1 e lnx x 1 0, x 0
x x
,
lnx x 1 lnx x 1 0
10
x x 0
0, .
F 1 2
1
F 2F
2
2
F 3F
3
.
F F 1 2
x 0
1 2
F 2F
F x F 3x 2F 2x .
G(x) 2F(x) F(3 .
G( F x f x 0 F
0, f
0
.
G(2 F x F 3x 2F 2x .
G( G
Bolzano
G F .
G x 2F x 2f x 0 G
51
133. f :
f x f xu 2
1
1e lnu du e x ln x 1
u x .
f x 0 x .
x 2f x e x 1 , x .
0,1 f 4 .
g: g f x x 4 f g x g x 4
x g
1,2 2f 5e 2e .
134. f
f x dx f x dx .
135. f f x 2 x
2
x 5x2
0g x x 5x 1 f t dt
, x
g 3 g 0 0 .
g x 0 3,0 .
136. f, 2 2
f t dt 2 f t dt
2
f t dt 0 .
137. g,h , g x h x 0
t
tg x h x dx g x h x dx 0 , .
138. f : 0,1 3e 2 f x 3 3x
x 0,1
xe x 1 x .
x
x
0f t dt 3e 2 0,1 .
139. f,g
x
0f x x 1 g t dt x ,
3g x
4 x 0,4 g
0,4 .
52
0,4
0 g t dt 1 .
0,
0g g t dt 0 .
140. f 0,2 2
0f t dt 2
0,2
0 f t dt 2 .
1 2, 0,2
1 2f f 1.
141. f,g, :
x x
1 4f t dt u du
x
21
4g tx dt, x
g t 4
f f x 1 x .
f f , .
f x x 1,1
16
4
tf dt 0
4.
142.
2 2x
3x
u 1g x du, x 0
u u.
g 0, g x .
g
22 2x x
3 31 1
u 1 u 1du du
u u u u x 0
g x 0 .
2 24 9
3 32 3
x 1 x 1dx dx
x x x x
1,2
2 2
2
3
x 1dx 2
x x.
1
2,3
21
1
2 2 2
3 3 3
x 1 x 1 x 12 dx dx dx
x x x x x x
143. 0,
1
x
1
1f x f t dt , 0 48 , x 0
x.
)
1
xx
1 1g x f t dt f t dt x 5
x
0, .
1
xx
21 1f t dt f t dt 2 x
x
53
1,2
1
5 2
1 1f t dt f t dt 2 .
144. f : 1,2 f 1 1 f 2 2 .
fC y x .
0
x 1,2 0 0
f x 3 x ,
1 2, 1,2
1 2 1 2
f f 1,
f 1,2 f
2
1xf x dx 0 xf x f x 1 1,2 .
145. f,g
x x
1 0f t dt x g t dt 1 g x 0 x
f 0
x 0 f 0 2g 0 .
g.
0
x 0,1 0
2x
0 00
g t dt 2f x 1 4x .
146. f x
2
1xf t dt x 2 x
x
1
0f t dt 2
f y 2
f 2 0 3
2f t dt 0
0x 2,3
0f x 0 .
147. f 9 9 .
.
.
f :
i. ii. iii. 9
0
81f x dx
2
148. f x f x
2f x e
x f 0 0 .
x1 e
f x ln2
f : 0,9 f 0 0
0
x 0,9 0f x 3
1 2
x ,x 0,9
1 2
1 23
f x f x
0,9
f 3 3 f x xf x 0
54
x
o
x 0
f x t dt
limx
.
x
2005
x
h x t f t dt 2007
xg x
2007 h x g x
x .
x
2005
x
1t f t dt
2008 0,1 .
55
olzano
149. z1,z
2
2 2 2
1 2 1 2 1 2z z z z Re z z 0
f : , ,
2 2z if ,w f i 0
2 2 2w z w z
f x 0 , .
2 2 2
1 2 1 2 1 1 2 2 1 2 1 2z z z z z z z z z z z z
1 1 2 2 1 1 1 2 1 2 2 2 1 2 1 2 1 2 1 2
z z z z z z z z z z z z z z z z 0 2Re z z 0 Re z z 0 .
2 2 2
w z w z Re wz 0
2 2 2 2 2 2w z f i if f if f i f
2 2 2 2f f i f f .
2
2 2
2Re wz 0 f f 0 f f .
2
2f f f 0 .
- f f 0 f , Bolzano
f x 0 , .
- f f 0 f 0 f 0 f x 0 .
f f x 0 , .
150.
2 2
22
x z x zf x
x z, z i , , , 0 .
x xlim f x lim f x .
f z 1 z 1
f.
2 2
2 22 2
x z x z (x z) x z x z (x z)f x
x z x z
2 2
2 22 2
2 z z xx z x z x z z x z x z x z zf x f x
x z x z.
z z 2Re z 2 0 2 2 2
z .
56
f2 2 2
4 xf x , A
x.
2 2 2 2x x x x
4 x 4 x 4lim f x lim lim lim 0
xx x
xlim f(x) 0 .
2 2
z 1 z 1 z 1 z 1 (z 1) z 1 (z 1) z 1
z z z z 1 z z z z 1 2 z z 0 2 2 0 0 .
f
2 2 2 2 2 2
2 22 2 2 2 2 2
4 x 4 x 2x 4 xf x
x x
22 2 2 2 2 2 2
f x 0 4 x 0 x 0 x z x z
x z x z .
x z f x 0 f , z1 .
x z , z f x 0 f z , z2
x z f x 0 f z ,1
f x z
2 2 2 2 2
4 z 4 z 2 2f z
zz z 2 z
2 2
2f z .
1, z
12 2x
2f lim f(x),f z 0, .
2z , z
2
2 2 2 2
2 2f f z ,f z , .
3z ,
32 2x
2f f z , lim f(x) ,0 .
f
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2f(A) 0, , ,0 , .
2
f 0 2
x 0f x 0 .
f 2 0
x f 2.
1
0 f 2
0 f f x 0 0
x .
151. f : , f x 0 x ,
z Re z 0, Im z 0 Re z Im z 1
z fz
2 2
2
1z f
z
z 1
57
2 2f f
3x f f 0 1,1 .
1
z fz
1Im z 0
z.
*x y z x yi,
2 2 2 2 2 2
1 1 x yi x yz x yi x yi x y i
z x yi x y x y x y.
y 0
2 2 2 2 2 2
1 y y 1Im z 0 y 0 y 1
z x y x y x y
22 2
x y 1 z 1 z 1.
1
z fz
2
2 2 2 2 2
2
1 1z f z 2 f f ( ) 2 f
z z 2 2 2
f f 2 f .
3
g x x f f , x 1,1 .
g 1,1
g 1 f f g 1 f f .
2 2
g 1 g 1 f f f f f f 0 .
Bolzano 3g x 0 x f f 0
1,1 .
152. z K 3, 4
5 .
2 2z x 4x x 4x 1 i 2x
1,3 .
2
x 3
z x x 1 i 25lim 12 2Re z
x 3z.
z 10 .
z K 3, 4
z 3 4i 5 .
2 2f x z x 4x x 4x 1 i 2x , x 1,3 . f
1,3
58
f 1 z 1 4 1 4 1 i 2 z 3 4i 2 5 2 3 0 ,
f 3 z 9 12 9 12 1 i 6 z 3 4i 6 5 6 1 0 ,
f 1 f 3 0 Bolzano f x 0
2 2z x 4x x 4x 1 i 2x 1,3 .
z i , ,
2
x 3
z x x 1 i 25lim 12 2Re z
x 3
2
x 3
i x x 1 i 25lim 12 2Re z
x 3
2
x 3
x x 1 i 25lim 12 2Re z
x 3
2 2
x 3
x x 1 25lim 12 2Re z
x 3
2 2 2 2
x 3
2 x x x 1 2 x 2 2x 25lim 12 2Re z
x 3
2 2 2
x 3
2x 2 1 x 2 24lim 12 2Re z
x 3 1
2 2
z 3 4i 5 3 4 25
2 26 9 8 16 25 2 2
6 8 2 .
1
2
x 3
2x 2 1 x 6 8 2 24lim 12 2Re z
x 3
2
x 3
2x 2 1 x 6 6 24lim 12 2Re z
x 3
x 3
x 3lim
2x 2 8 2
x 3 12 2Re z
x 3lim 2x 2 8 2 12 2Re z
6 2 8 2 12 2 2 2 1.
2 2 21 6 8 6 7 0 1 .
1 1 z 1 i 7 1 z 7 i .
22
z 3 4i z 3 4i z 3 4i 5 z 3 4 z 5 5
5 z 5 5 0 z 10
153. f : 0,1 0,
z x x 1 i , w f x i , x 0,1
2 2 1 6 6 24 3
6 6 6 24
2 2 2 8 0
59
x 0,1
z w z w 2
i. 100w z ii.
100 100
z w z w iii. 100 100
z kw kz w , k
z w z w 2
i. 4 4z w , ,
ii u, wu, zu, u
2 2 2 2 2 2
OA OB AB z w z w
22 22 2
x x 1 f x 1 f x x 2 x xf x x 1 0 .
g x xf x x 1 g 0 g 1 0 Bolzano
22 2 2 2
z w x x 1 f x 1 f x 2x 2x 1
x 0,1 22x 2x 0 2
f x 0 1
2
2
x 02x x 1 02x 2x 0
f x 0f x 0f x 0 f 0 0 f 1 0
f 0 0 z i , w i z w 2 2 .
f 1 0 z 1, w i z w 2 .
z 1 w i
i. 25
100 100 4w i i 1 z
ii.
50 50100 100 100 100 2 2
z w z w 1 i 1 i 1 i 1 i
50 50 50 50 50 50
2i 2i 2 i 2 i
iii. 100 100100 100 100 1002 100
z kw 1 ki i ki i k i i kz w kz w
z w z w 2 z i , w i
i.
4 44 4 4 4z w i i i i 1 1
ii. u,wu, zu
AB u wu u 1 i u 2 , A u wu u 1 i u 2
B zu wu 2iu 2 u .
AB A
2 2 2
AB A B
60
154. f : f x
3x
1
1g x z f t dt 3 z x 1 0
z z i ,
,
g g .
1
z zz
2 1Re z
2.
f 2 0 , f 3
0
x 2,3 0f x 0 .
f z f
x
1
h(x) z f(t)dt
3x
3
1
(x) h x z f(t)dt
1
g(x) (x) 3 z (x 1)z
3 2 1g (x) z f x 3x 3 z
z.
g 1 0 g x g 1 g 0
x 1
g
Fermat
1 1 1
g 1 0 z f 1 3 3 z 0 3 z 3 z z zz z z
.
2
21 1 1 1z z z z zz z z
z z z z
2 2 2 2z z 1 z z 1zz zz 0 z z 1 0 z z 1
z z zz z z zz
2 2 12Re z 1 Re z
2.
z i 2 2 2z 2 i 2 2 2
Re z .
2 2 10 ( )( ) 0
2.
0 0 0 0 .
f 2,3 f 2 0 f 3 0
f 2 f 3 0 Bolzano 0 0
x (2,3) : f x 0 .
155. xw e x 1 i , x .
0,1 2z w w 2i
0w
61
4k4k
0 0w w , k .
k 4k
0w
2
2 x xz w w 2i e x 1 i e x 1 i 2i
2x x xz e 2e x 1 i x 1 e x 1 i 2i 2x x x
z e e x 1 2e x 1 x 1 i
xf x 2e x 1 x 1, x 0,1 . f 0 2 1 1 0 , f 1 2 0
f 0 f 1 0 . f 0,1 Bolzano
0,1 f 0 0,1 Im z 0 z .
22x
w e x 1 22x
g x e x 1 , x . g
2x22x
2 22x 2x
1 e x 1g x e x 1
2 e x 1 e x 1
.
g
2xe x 1 2x
h x e x 1, x .
E 2xh x 2e 1 0 h .
x 0 h x h 0 0 g x 0 g 0, .
x 0 h x h 0 0 g x 0 g ,0 .
g x 0 g 0 2 w 0
0w e 0 1 i 1 i .
4k4k
0 0w w
2k 2k4k 4k 2 2
1 i 1 i 1 i 1 i
2k 2k 2k 2k 2k 2k
2i 2i 2 i 2 i
2k k4k 2 k4k 2k 2k 2k 2 2k
0w 1 i 1 i 2 i 2 i 2 1 .
4k
0w
k 2t , t .
mint 1
mink 4 .
62
156. f : 3f x 3f x x 3
x .
1
f .
.
1f x f x .
1f f z 6 8i 1 3
z.
3 z 17 .
z i , , 2 24
6 x 8 x 5 0
0,1 .
157. 2 22
f x 3 x 4 x 1 ,
z i K 3, 4
2
f x 0 0,1 .
2 z 1 2i 3 2 .
z 4 5i
2x
f xlim 1
x 2x 4.
158. x z * 2 33
f(x) x z x z , .
f .
f.
f x 0 0, z .
3
x 0
f x zlim 1
xz.
159. f z x f x xi , x .
z z 1 z z
f 21 .
z x 2
i. f x 0
ii. f 2,2 .
iii. f 0 2 f.
iv.
2
x 0
f x f x 6lim
x
63
160.
2
xz w , x 1f x
z x x w , x 1z,w
w 0 .
z
uw
z f i , w f i , , 0
f x 0 , .
w 4 f 2 2f 1
z.
161. f z 1 f 1 i
z 9i 3 z i
z 3
2f 1 8
0
x 0,1 2
0 0 02x f x 5 4x .
w 3w iz 2
w.
