f,g f f. f - askisopolis.gr · f x 3 4 2 5 xx ... 3f 101 2f 1001 f 10001 log18 6 28. f x ln e 1 x g...
TRANSCRIPT
1
1. 2f x 2 .
2
2 2f x 2 x 8 f x 2 x 20 0
2. f x 2 2 x g x x , , 0 .
f,g
1.
f g3 4
f x 3 2g x ,2 .
3. f x 2 2x , x .
f f.
) f
3
x 0, , , ,4 2 4
f 0 x .
) f 5
y2
.
2
f x 2 2f x 3 0
4. f x x
,34
.
f.
f 0,2 .
f x f x3 6
5.
1 1f x x 1 x 1
x x.
f x 2 0, .
6.
f x 2x 2x2
f x 2 2x .
f
2
f
N 0, f x 1 0 .
7.
x 20 x 22
f x7
x 5 x 42
.
f.
2 .
1
f x2
.
8.
x 1f x
x
f x x
9. 2f x x 2 x 2 , x
f x 0 , x .
f x 0 .
f
f
f
10. 2f x 3 x ( 3 1) x 1 ,
x 0,2
.
f x 0 .
92
21
171821
2
11.
1 2 x xf x
x x.
f x x x .
f x f x 1 0 .
3
12. f x 2 x .
2 2f x x 3 x .
2 2
3 9f 8 x f x f x f 7 x
2 2
13f x f x
2
x.
13.
x xf x
1 x.
1f x 1
x.
f
4k 1f x f x
2 2
f x 3
2.
14. 3 2f x 2 2x 2x 4 2x 2 .
3f x f x
2 2.
f x 0 .
3g x 2 2x 2 2x f x
15.
x 12
x 1
16.
2
2
16 xf x
1 x 2
g x x 1.
f x 3g x .
2f x 16 x
2
1g x
x.
2 216 x x 3 0 .
17.
3f x 2 4x 4x 1
2.
f x 4x .
f.
4
4 22f x 5 4x 2 0 .
18. 2x
f x , 0,3
y y
2 1.
f
2
2 3f x 1 f x 1 4
4.
f 6x f 3x 0, .
19.
f x 2x4
g x 2x4
.
3f f f f 0
2 4 4.
2 2f x g x 1.
20.
3f x 2x 2 2 2x
2.
f x 3 2x .
f
0,2 .
3
f x2
0,
2
2 2f x f x 9
4
21.
t
f t 20 43
t t 0,24 .
f.
m
m
22.
f x x4
, ,
A , 1 , B ,14 4
) 2 1
5
)
) f .
)
f 2x 24
.
3 5 7 9f f f f f 1
4 4 4 4 4
23. 2f x x x 2, x .
f 2 .
f
f 0,
0,2
f x f x2
.
24. 2 2f x x x 3 x 2, x .
f 2 .
f 1
8.
f
) f
g x 2 3 x .
) 0,
f x f x2
.
25. x x x xf x 36 225 3 4 3 25 .
f 0 5 f 0 3 f 0 40 f 0 1052log log log log 0
f 0 2 f 0 11 f 0 77 f 0 32
26. x x x x3 4 2 5 4 3 5 2
x xf x 3 4 2 5 x x
g x 4 3 5 2
f
27. f x log log x 1 .
f.
f
f x 7 f x 1 log2 .
3f 101 2f 1001 f 10001 log18
6
28. xf x ln e 1 1 x
g x ln 2e 2 .
f,g.
f,g
f 2x g x 1 .
g 1 x f x ln2
29. 2f x log x 1 log x 2x g x 1 log x 1 .
f,g.
i. f x 0 ii. g x 0 .
f 4 3g 1 4 log30 .
30. 2f x log 2x 4x 4 2logx .
f
2x 1 f 1 xf 1 f 1
10 5 10 10 f 2
f 1 logxx 2
2f 1 logxx 2 2
31.
x 1f x ln x
1 x
f.
f
1 1f , f , f 0
2 2.
f x f x 1 1 2x 0 .
32. 4 2
f x logx 8 logx log 100x ,x 0 ,
f
M 10,25 .
i. f 2
2f x log x 4logx
ii. f x 0 .
33.
2x
x
e 1f x ln
e 5.
) ) f x 2ln2 .
) f x 0 .
f xxg x e 5 e .
i. g -1. ii. g x x g x
7
34. 2x xf x ln e 2e 3 x
g x ln3 ln e 1 .
) f g.
) f x g x
) f x 2g x .
g 1 1.
g x g 1 f 0
35.
2x x
1 1f x 2 3 1
5 5
xg x 5 .
49125 5 1
g x g x 1 g x 2 ..... g x 494
36. 2x xf x ln e 4e 3
x x .
.
37. 2x x
x
f x x
f x x 2ln2
f -
2x 1 x2x 1 e x e
38. 2x xf x ln e 6e 8 .
.
f x ln3
f x x ln3
39. x
3f x 5 .
x 1 x 3 x 4 x 2f 7 3 5 f 3 5 .
f
A 1,19 .
3 f x f x 2
8
40. x x1f x e e
2 x x1
g x e e2
.
f g
2 2f x g x 1 x .
g g f g f , , .
f x 2g x 1.
41. 2f x logx .
f
g x f x 2
y 2log4 .
42. f x log x 4 3 .
f
k k, 2 f.
x 3 x x x 3f 3 2 f 23 3 2
43.
log 3x 14f x
log x 4.
f y 2 .
f y 1.
44. f x ln x 2 .
f
f
) 2ln x 2 f x 2 0 .
45. f x log log x 4 .
f
f x 0 .
46. x
2f x 4 4
f
1,9 .
2,3 2f x f 5x 6 .
4 x x2f x 3 9 5 6 0 .
9
47.
lnx 1f x
lnx 1.
f
1f x
1f
x
1 1x 0, ,e e,
e e.
1f x 1
1f
x
48.
x
4f x
2
6 7f f
7 6.
.
0 .
i. f x 1 f x 2 f x 3 48
ii. log2
2f logx log f x
iii. logx log22 x 16
49.
2f x x 4x 3 .
g x f x 2x .
h x f x 4x
i. 2h x 3h x 4 ii. h x x 1
50.
3 2 2P x 4x 3 x x 2, ,
2.
P x x
2
P x 2
6 2P 1 11 0 .
51. x
f x log 2, x 0 .
f x 2 .
i.
2 f 10
10 50 ii.
1f 2 f 0
2
x
y
1 2 3
-1
10
52.
2
3
5 1x x
3 3
x 1
18 , x 0,2
4.
4
3.
43
1 3log
3 ln2 43
2
6 27A log e
4
53. x
f x 2 , 0,2
f
f 2 f 1 6 .
54. f x 2 2x 1, x .
) f
)
f x f x4
0, .
)
5f log2 log f 3log f log128
4 12 12.
) , 3 2P x x 6x x
x f
4
x f
12
55.
2
2
2 2x 1f x
2 x 1.
f x 0 .
log x f log x f log36 6
56. 3 2P x x x x 1 x 1
x 1
2 7 .
x 0 .
x ,2
3 22 7 2 3 0 .
57. 3 2P x 2x 7x x x 1
x 1 18 .
7 2 .
3 22 2x 7 2x 7 2x 2 0 .
11
x x x2 8 7 4 7 2 2 0 .
58.
1
4 2 1 3 22f x x 5 x 2 3 x 25 x 1
x 1) f x
1
2
i. 2
x 1 f x .
ii. f
x x.
59. x xf x log 2 4 5 2 2 .
f.
f
y log2 .
0,2 f 4 x f 2
60. 2
f x logx log x .
f.
f
f x 0 .
f f
10.000 .
61. x xf x 9 3 12 .
x x f 1
3x 7x 6 f 1 .
62.
f x ln x k3
.
ef ln ln
6 2k.
k 1
i.
0,
2
ln x f 2ln2 f ln3
6.
ii.
f x 1 f x 1
e e .
63. f x 1 x lnx, x 0
f
x x A 1,0 .
12
x 2x 5f e 1 e x
0 x 1 f x 0 .
0,
2
eln .
64. 3 2P x x 3x logk 4 x logk ,k 0 .
x 1 P x .
k P x
k 1000 :
i. xP e 0 ii. 2
P x x
65. 3 2P x x x x 4 , ,
P x 2x x 2x 4
2 1.
P x x 1
N x 2xP 10 10 5 .
1 3log x P 1 logx log
2 P 0
66. 3P x x x 2 4 , ,
2
x 1 .
3 1
P x 0
P x 0
1
2 log 3P 2 2P 0210 25
67.
x y 12
3 x 2 y 8
,
, 0,2
.
, 4, 2
68.
2
x y 2
.
0 0(x ,y ) .
2
0 0 .
13
69. k 1 x 2y k 1
kx ky 1
k
0 0x ,y
k 1,
i. 2
0 0y x 0 .
ii. 0 0x t t yt 3 t 4
021 3 x 5 3 5
70. x y
D, D , D
x y
D D 4D x y
D D 2D
0 0
x ,y 3, 1 .
0 0x y
5, 0
x
5.
0 0 0
log32 log x y x log2 0 .
71. f x log logx .
f x 1.
x
g x f
.
i.
ii. 3g x 4x g x 4
72. 2f x ln x 1 x .
f
73. 4 3 2P x x x 2x x , ,
f x logP x . P x 2
x 1
2 1.
P 2 x 0 .
f.
f 7 f 2 f 3 log9 .
74. 3 2f x 2x 5x x , ,
2
x 1 .
3 .
4 1.
f
14
3 22 x 5 x 4 x 1
x f 0 y 2
x y 2 f 1.
i. 0 0x ,y .
ii. 0
x
0
y .
75. 2 2P x 1 x 1 x 1 .
P x 1 2
x ,x
1 2 1 2
x x x x 1.
3 2Q x x x x , , P x
Q x
76. xf x x ln e 3 .
f
f 1821 f 2015 .
f
y ln10 .
) 2
2 x 3 4x ln e 3 1 ln e 3
77.
xln ln3
f xln ln
.
ln ln .
23 .
6 24
i.
x1
f x2
ii. ,
2 f f f2
iii. 2 2f x f x 3