g lukeiou papagrigorakis 2016 17
TRANSCRIPT
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2016 - 2017
M
333
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: - &
: - 16.08
M.ed. 2016
: http://users.sch.gr/mipapagr
email: [email protected]
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3
. 2016-2017
1 1.01 2
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x
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1.07 N
f(x) ln x
6f x 10
1.08
f x x g x x 1 h x 1 x
1k xx
1m xx 1
1n xx 1
1.09
f(x) ln( x), x 0 g(x) ln( x), x 0
k(x) ln x m(x) ln x t(x) ln x
1.10 N
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) t(x) 2 2x ) 2f(x) x
1.11 :
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f(x)3 x , x 1
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2
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5
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1.21
2f x x ln x 2(x) x x x 1g(x)
x x-3x+2 x
t(x) 1
x 2 x 1
h(x) 2
x 2
1 x 1 x
24 - xk(x)(x - 1) x 1
1.22 :
2 xf x2x 1
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21t x
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2x 1 ln x
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p(x) xe - 1 + 1 - lnx 2q x ln 1 x
1.23 R
2f(x) ln x 4x 1 R
1.24
2
x x 1x x 1f x
9 4.3 27
x 2
1h x4 x
k(x) 2x 1 xm(x) (e 1)ln(x 1)
1t x ln xx 1
1r x ln xx 1
3 2p(x) x x 1
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1.25
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1.27
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x 2, 5
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1.28 :
2x 2 2 x 3f(x)x 1 3 x 5
, g(x) 3 2 x 1
1.29
f(x) 1log 1x
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x 15 eg x
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1.30 y f x .
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1.31 f(x) x 1 .
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f . 2
1x - 1f (x)x - 1
3
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x - x 1
23f (x) x 1 4 1f (x) x 1x
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.
1.32
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1.33 E
x xf x 1 2 2 1 g x 0
1.34 E
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1.35 .
) ln xf(x) e ln xg(x) e
) f(x) 1ln 2x
g x ln 1 2x ln x
1.36 R
3
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x x 4
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1.37 f,g : R R
x R
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1.38 f g , gf
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1.39 f g , gf
f(x) 2x 1, x 2
x , x 2
lnx, 0 x 3g x
-2x 3, x 3
1.40 f,g : R R
2 2g x f x 2f x x 3 , x R .
gC Oy
1.41 N f : R R
: 2 2f x x 1 , x R
1.42 N f g ,
2 2f g (x) 2 f g (x) 2 x R
1.43
f : R R 2 2f (x) 2f x x
x R f x 1 x R
1.44 f : R R x R
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1.45 f : R R
: f x x , x R
1.46 f : R R
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1.47 f x, y R f(x)f(y) 1 f(x) f(y) xy
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7
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1.48 N
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1.49 f : R R x f(x) f( x) 2 2f( x) 0, x R .
f .
1.50 ** f : R R f(x y) f(x y) 2f(x) f(y)
x, y R . : ) f 0 0
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1.51 f : R R
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1.52 f,g : R R
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x R . f
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1.53 f(x y) f(x) f(y) , x, y R
f
1.54 f : R R f(x) 0 x R
1.55 f : R R g(x) f(x) + f( x)
1.56 f,g
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1.57 f ( )
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2 2f(x) ln x 1 ln x 3 xf(x) x
1.58 2f x 1 x , g x 3 x 2
f g g f
1.59 f g
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) f(x) x 1 x (0, 2)x 1 x [2, 4)
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1.60 f(x) 2ln(x x 1)
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1.61
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1.62 f :
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1.63 f : R R
(1 x)f(x 1) f(1 x) x 1 , x R
1.64 ff : A R ,
gg : A R f gf A A . :
A) f , gof .
B) f , g f
.
1.65 f(x) f(2 x) x , x R
1.66 B f : 0, R
xf ln x f x 1e
x 0 .
1.67 f
2f x x x 1 f x 1 x x R
1.68 2xf f(x) e x R ,
f 2014
1.69 f f (x) 2x 1
x R f 1
1.70 f : ) (1 x)f(x 1) f(1 x) x 1 , x R
) 212f(x) f xx
, x R *
1.71 f(x) x 32 x
R ,
: f f (x) x x 2
1.72 f : R R
f f(x) 2x 1 , x R .
f x 1
1.73 * *f : R R
f(2004) 1 x, y 0
2004 2004f(x)f(y) f f 2f(xy)x y
,
1.74 *f : R R f(y ) 1f(x y) f(x) e x, y R .
f(x) 1f(x) e 1 f( x)f(x) e x R
f
1.75 f : R R
x ye f x f y f x y , x, y R .
f 0 1 , 1f x
f x , x R
f
1.76 f : R R
2f x 2 f 3x 0 , x R . fC x x
.
1.77 f : 0, R
xf x f y lny
x, y 0,
f 1 0 . f x ln x , x 0
1.78
f : R 0 R 1 1f x f xx x
x R 0
1.79 ** f : R R ,
f(x) f( x) x 3
x ,1 f(x) f( x) x 3
x 0, . A f(3) 4, f( 3) 2 ,
f(x) (www.mathematica.gr)
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9
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1.80
f(x) 5 5 x xg x ln e x
3t x x 1 2 4 xm x e 3
r(x) x 3x 2
(x) ln x 0 x 2
1 2x x 2
1.81
k x ln x , g(x) 2x 1
x 1
m x ln x
1.82 f : 0, 0,
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f x x
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1.86
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5 5
, x R
f . .
) x x x3 4 5 .
1.88 f : (0, ) R
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x, y 0 .
1 f 0 .
f . 0,
1.89 f,g
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g x f x , x , f
. f g(x) g f(x) x ,
1.90 f : R R R f(f(x )) x x R ,
f(x) x , x R
1.91 f : R R ,
2 22f (x ) 2xf(6x 8) 4 3x, x R
1.92 f : 0, 0,
31 f x 1 x xf x
x 0, . f
1.93 * f
x R : 2x 3f(x)f x
5
,
f(x) x , x R
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1.94 f : R R R .
5 2 2 5f x x 1 f 1 x x 2 x x
1.95 N :
) ln x 1 x ) x 1 xe1 x
1.96 3 xf x x 5 1
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7x 12 3x 183 3
7x 12 3x 18
5 5(3x 18) (7x 12)5 1 5 1
..
1.97 f : R R ,
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) 2f x 2 f x 2014
1.98 f : R R
f f(x) 3x 0 x R .
1.99 f : R R ,
g x f f(x) 1,0
(mathematica.gr)
1.100 f 0, .
2 3f x f x f x f x
1.101 f : 0, R
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g x f h(x)
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x 2x 11x 111xh e h e h e h e 1,1
1.102 * f : R R
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1.103 f(x) ln x x 1
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f x ln x x 1
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1.105 f(x) x ln x ,
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) x 1ln 1 2 x x2 x
1.106 * f : R R
f x xe f x 1 e
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1.107
g(x) 4 x 2 43t(x) 4 x 4x 2r x x 4x 5 f : [ 1, 4) R f(x) 2x 1
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1.108
) f x 1 2ln x 1 , x 2, 3
) f : [ 1, 4) R f(x) 2x 1
1.109
) 2f x x 4x 5
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1.112 f : R R f(0) 1
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1.113 R , 2f(x) x ( 1)x 2 2
1.114 f,g : R R
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h x f x f 1 x g x g 1 x
(mathematica)
1.115 2f x x 6x 8 , x R ,
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1.116 2f x x 6x 8 , x R ,
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23 2 2x 8x 85 x 4x 4 e
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1:1
1.119 , 1 1 :
) x 1f x 3e 2 ) xf x e x 1
) f x x ln x 2 ) 5 3f x x x x
1.120 f : R R 2015f f x 3f x x 0 , x R
1-1
1.121 f : [1, ) R
2f(f(x)) 2x 3x 2 ,
x [1, ) . f 1 1
1.122 f : R R 1 1 . 3F x f x 2f x 3
1 1 .
1.123 R 1 1
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x 8 x 0
1.124 1 1
f 2 26f x f (x) 9 x R
1.125 f x 2 x ln x
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1.126 .
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1.127 N
442 2log 1 log 5 5 5 5 1
1.128 f,g : R R
2f f (x) x 5x 9 2g x x xf x 3 ,
x R . N f 3 3 g
1 1
1.129 2f x 2x ln x 1 , x 0 f 1-1
:
2
24
3x - 2 +12 x - 3x +2 = ln
x +1
2,
1.130 f : R g : R , B f(A)
g f 1 1 g 1 1
1.131 yxx e y e , x, y R
) x y .
) 22 3x x 2x 3x 2 e e
1.132 3 3e e , R
1.133 x
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1.134 x 3f x e x x 1
) N 1 1
) : 2x -x 2 3 2 x+3 3e +(x - x) + x - 2x = e +(x +3) +3
1.135 f : R R :
52f (x) f x 3x x R .
) f
) 2f x x 1 1
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13
. 2016-2017
1.136 f : R R
f xf x 2e x 2 x R
) f
)
xg x x 2e
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1.137 f R ,
31 2f x 1 x
f x x R .
) f
f 0
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1.138 f , R
3f x 3 xf x e x e , x R .
f 3f x x
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5 3h x x x x , x R .
) f R
5 3f x f x f x x
x R . f
) h x 3
f 3
1.140 f R , : f 1 f(x) 2x 6 f x
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:
( 1)
1.141 f : (0, ) R
: xf x -f y =fy
x, y 0
f x 0 ,
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2 2f x f x 3 f x 1 f x 1
1.142 f : R R
: x xf(e x) 8f(x 1) 2008 e
x R . f
: : f x 223
x xf e 2 e 223 .
1.143 f : R R
f xe f x x x R
) f 1-1
) f 1 .
) x 4 2x 1e e x 5
1.144 *** 1 1
f : 0,1 R f 0 f 1 1 .
1 2x , x 0,1
1 24f x 2f x 1 .
1.145 f
0, f (x )f e lnx
x 0, . f x ln x ,
x 0,
1.146 f : 0, R
f(x)f e lnx . f .
x R
f 1 1 f(3) 2
2f(1 2f(x x 1)) f(1 f(5)) 4
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1.147
) 3f(x) x 1 ) f(x) 5 x 2
) xf(x) log 3 10 ) xf(x) ln(2 e ) x
) 2f x 2 x 3 , x 3
1.148
) f(x) 2x 3 .x 4
) f(x) 3x
3x3 e3 e
) xf(x) log1 x
) xf(x)1 x
) 2x 1 , x 0
f(x)9x , x 0
) 3 2f(x) x 3x 3x ) 2f(x) ln x 1 x
1.149 1fC fC
f(x) 1 x , x 1,0
1.150 f f(f(f(x))) 2x 7 x R .
f(1) 3, f(3) 9 . f
1 1 1f (x) 9 .
1.151 f 3f(x) 2x x 2 .
) f .
) 1f(x) f (x) .
) 1f (5x 6) 1 .
1.152 f x x ln x
) 1f e 1
) 2
22
2 1ln 4 5
1.153 f , g : R R ,
1f g 1g f ,
1g 1f
1.154 3f(x) x x 2
)
) f(x) 12 , 1f (x) 2
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y x
) 2 3 3 2(2 x) x x x 2
: 1f (x) 3 , 1f (x 1) x 5
1.155 f : R R f R R
3f (x) 3f(x) x 0 , x R .
f , 1f
fC 1fC
1.156 f,g : R R
R
f g g f , 1 1f g g f
1.157
y x
1.158 f : R R 5f(x) x x 1 .
A) 11 13f f
B) 1x f (x)
1.159
5x 2f x2x 5
y x
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1.160 f : R R
f f f ... x ... 2x 1
f 1
1.161 1 1 f : R (0, ) f(x y) f(x) f(y)
x, y R . : 1 1 1f (xy) f (x) f (y) , x, y f(R)
1.162 f : R R 1, x R 2 2xf (x) 2f(x) e 1 . f .
1.163 f : R R f x y f x f y , x, y R .
x R : x 0 f x 0 .
) f
) 2 2f 4x 2005 f 4x 2005 2f 8x 4
1.164 x
xe 1f xe 1
1 xg x ln
1 x
.
f 1f g
g f
g
f , 1f
: f 1 1 1f .
2 2x g x x g x
1.165 ) f . R ox R , o o o of f(x x f x x
B) 34x 1f x3
, 1f
fC 1fC .
1.166 f : R R f(x y) f(x)f(y) x, y R R ,
f() 0 . :
) f(x) 0 x R f(0) 1 ) f( x) = 1f(x)
f(x)f(x y)f(y)
, x R
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1.167 H f : R R , R fC
A 5,9 B 2, 3 :
) f .
) 1 2f 3 f (x 2x) 9 1 2f x ln 1 2x
) 2f x 12f x 27 f x ln x 4 9
1.168 3
3 3x 1 2x 12 3
1.169 f : 0, 0, f 1 1 g x xf x 1
.
) f .
B) N f x ln e x 0
) 7 5 9f x f x f x f x
1.170 f : R R : f(x)f(x) e x , x R
f ..
fC x x , 1
: f(x) x x R
: f(x)2f(2x 1) e x .
1.171 f : R g : B R ,
)
)
)
1.172 ) f, g R . :
) g R g f R , f
R
) g R f g R , f
R
) f ,g x
xe 1f x
e
g x ln x 1
) g f g f(x) x x R
) 2 2x 1 x 1 x 1 x 1e 1 e e 1 e
f(A) B g f 1 1 f 1 1
B f(A) g f 1 1 g 1 1
f(A) B g f 1 1 f g 1 1
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. 2016-2017
X0
1.173
) 2 3x 12 3lim
x 1 x 1
) 1
x 1
x ( 1)x 1lim *x 1
1.174
) x 2lim
3 6 x x 6x 2
) 3 2x 0lim x x
) 2
x 9
x 81lim2x x 6x 3 x 9
1.175
) x 1lim
3 2 3x 2 x 1x 2 x 1
) x 1lim
2x 3 x 4x 3x 1
) x 1lim
2x 1 x 1x 1
, )
x 1
x 1 x 1lim
2 x 1
1.176 x 0lim f(x)
x 2 x 2 x 0f(x) 4x
0 x 0
1.177 :
) x 0
6x 3xlimx 2x
) 2
1lim
) 2 2
2x 0
x 2 x 2limx 4 2
1.178
) x 1
x 1lim 1x 1 x
) x 1
1 x 1lim 1 x 2
1.179 :
x 3
xlim
x 1 2
x 0
2x xlim
x 1 x
1.180 :
)
x 1
x 3 2lim
(x 1)
)
22x 0
xlim
x
) 2x 0
1 1 xlim
x
1.181 ( )
x 0lim
1x x
2
2x 0
1x xlim
x x
1.182 N
x 0
x 2x ... xlim 28x
1.183 x 1
f(x) 5lim 0f(x) 2
x 1lim f(x) 5
1.184 x 0lim
f(x) 2x
x 0lim 2 2
xf(3x)-f(-x) 2x3x x
1.185 x 2lim g(x) 7
,
2
2x 2
2 g(x) g(x) x xf(x)lim
x x 4
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1.186 x 1lim f(x)g(x)
x 1 x 1
g(x)lim f(x)(x 1) lim 3
x 1
1.187 2xlim f x 0
,
0xlim f x
.
1.188 f R
x 3lim f(x) 2x 5 4
.
x 3lim f(x)
1.189 2x 3lim 12f(x) 4f (x) 9
,
x 3limf(x)
1.190 x 2
f(x) 4lim 1x-2
f x f 1 x , x R x 1lim f(x)
1.191
1xlim f(x)
1x
lim g(x) ,
15
xlim f(x) g(x)
1
2 4xlim f(x) g(x)
1.192 f : R R
x 1lim
f(x) x 2x-1
,
x 1lim 2 2
f(x) 1f (x) x
x 1lim
2
2
f (x) f(x) 2
f (x) 3 2x
1.193 A 2 2f x 2f x x 0
x R x 0lim f(x)
=1.
1.194 f: R R
x 1lim f(x) 2
x 0lim[f(x-1)-f(1-x)]
1.195 x 3
f(x) xlim 2
x-3
, x 3lim f(x)
1.196 f x 0
f(x)lim 3x
) x 0
f(vx)lim 3vx
, v 0
) 2 2f (vx) x 2f(vx) x
x R 3v 1
1.197 f
f x y f x f y , x, y R .
) f
) x 0lim f x 0
,
x lim f x f
R
)
x 0
f xlim 2
x
x 2f x 2
lim 2x 2
x 0
f (x) f(x)lim 0
x
1.198 f : R R *
3 2 3f x 2x f x 3 x ,
x R * . x 0
f(x)lim Rx
,
1
x 0
f(x)limx
x 0
f(f(x))limx
2
2x 1
f(x x)limx 3x 2
.
1.199 *f : R R
: xf(x) f(y) fy
x, y 0
) f(x) 0
1 f 1 1
) x 1
f(x)lim 2x 1
x4
f(x) f(x)limx x
1.200 f
x 0
f(4x)lim 3f x
. N
x 0
f(64x)lim 3f x
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19
. 2016-2017
1.201 ( )
) x 4
2 xlimx 3 x 2
) 3
3
x 1
x 1lim(x 1)
) x 4
5 2xlimx x 2x 2 x 4
) 2
x
5 xlimx
1.202 ( )
) 2x 1
x 1lim
x 2 x 1
)
x 0
x 16 4limx x
) 2
x 1
x 5xlimx x x x 1
)
2
x 0
3x 2limx 1
) 2x 1x 5 x 3limx 1 (x 1)
)2
3x 0
3 2xlim1 x
1.203 A x 1
h(x)lim|x 2|
x 1limg(x)
1.204 2x 1lim (x 4)f(x) 3x 2
x 1lim f(x)
1.205 x 2
2x 3 5limf(x)
x 2lim f(x)
1.206 x 1limg(x) 3
x 1
g(x) 2x g(x) 6 4xlim
x x x x 1
1.207 2
2
x 2 , x 1x 1f(x)
x x 5 , x 1x 1
,, R
x 1lim f(x)
1.208 2
(x) x 0xf(x)
x x x 0x 2 x
x 0lim f(x)
R
1.209 22x 1 x
f(x) x x x
x 4lim f(x)
R
1.210 R 2 2x 9x 5lim
(x )
1.211 , R : 3 2
3x 1
x ( )x (2 1)x 3 lim Rx 3x 2
1.212 , R x 1
x x 2lim 8
x 3 2
1.213 R
2
3 2x -x 2f(x)
x -3x 3x-1
1 .
1.214 N R :
A) 2
2x 2
x x 6limx x
` B) 3x 1
x limx 7 2
1.215 N 2
x 4
x lim(x 4)( x 2)
, R
1.216 ,, R 3
x 2
x x 6lim 4x 2
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20 -
http://users.sch.gr/mipapagr
1.217
) xlim x x x x
) 2 2xlim x x 3 x
) 2 2xlim x x x 2 2x
1.218
) 2
x
x 2 xlimx x 2
) x x 1
x 2 x 3x
e 3lime 3
) 2
2x
x 2x 3 4lim
x 2x 5
) x 0
3 2 log xlim
1 2 log x
1.219
) 2 2x
x 2xlimx x 3
) 2
3x
xx xlimx
1.220 x
xx
ln(1 3 )limln(1 2 )
1.221 2
xlim x 1 x
1.222
2
t
ln(t t 1)limln t
1.223 f : 0, R
x
f
xlim 3
x
x
ln f xlim
ln x
1.224 o
2xx
x 1xlim e
1.225 3 2( 1)x ( )x x 3f(x)
x 1
xlim f(x)
, R
1.226 2x 2x 3f(x) -x-
x 1
, R xlim f(x) 3 11
1.227 2f(x) x 2x 3 x
xlim f(x)
R
1.228 2f(x) x 4 x
0 , , , x
3lim f(x)2
1.229 , R :
2xlim x 2x 3 x 12
1.230 0 ,
x -lim
x x
x x 2 1 -2 1
,
1.231 0 ,
x lim
x x 1
x 1 x 2 2
1.232 xlim f(x)
0 ,, R
2 2 2f(x) x 1 x 2 x 3
1.233 2 2x f x ln
x
0
x 0lim f x
, x lim f x
x lim f(x) ln(x)
.
-
21
. 2016-2017
1.234 f
: 2 2x f(x) x 2 , x 0 .
A) x 2
f(x)-4limx 2
B) x 2
f(x)-4limx 2 2
) 2
x 2
f (x)-16limx 2
) 2x 2f(x)-1 -3
limx 4
1.235 x 0lim f x
, x 0limg x
2 2f x g x 2f x 4g x 5 x , x R .
1.236
o
2 2x xlim f x g x 0 ,
ox x
lim f x
ox x
lim g x 0
1.237 A 2 2f x 2f x x 0
x R , x 0lim f(x) 1
1.238 f ox 2 x R
2x 2 f x x 7x 10 . x 2lim f(x)
1.239 2
2x 0
1x xlim
x x
1.240 ** f 2f x f x x x R . N
x
f x 1limx 2
1.241 2
x
x xlim 1x x
1.242 2
xlim x x 1 x
1.243 x
x xlimx x
1.244
xlim ln x 1 ln x x
1.245
2
x
x 3lim3 x x
1.246
2 2xx 2xlim
x x 3
1.247 x
x 1xxlim
x 1
1.248 x 2lim f(x) 4
f(x) 4
x R , :
2
2x 2
2f (x) 7f(x) 4lim
f (x) 16
x 2
f(x) 2lim
3f(x) 12
1.249 f R
xlim f(x) 5x-7 3
. :
xlim f(x)
, x
f(x)limx
, 2x2f(x)-3x 5lim5x 2x 1
1.250 f
x
f(x) xlimf(x) x
xlim f(x)
1.251 x
f(x)lim Rx
xlim f(x)-x
2f(x) 9x 1 3x
1.252 f : ,0 0,
x 2
xlim
1 x x f x
f x 0 , x R . Y xlim f x
1.253 f : 0, R
x
f xlim l 0,
x
.
x
ln f xlim
ln x
-
22 -
1.254 f
2xf(x) 5x 2 x (x 1)(x 2) , x R
1.255 x 2
f(x)-2xlim 1x-2
f
, 2
x 2
xf(x)-2x 3f(2)-6xlimx-2
1.256 x R
2 2 2 x 2xf(x) f (x) x x x 2f(x) .
f ox 0 .
1.257 f : R R
5f x f x x x R .
ox 0 .
1.258 H f 0
xxf(x) e 1 , x R . B f(0)
1.259 1f x ln xx
1f
,
1 2
1x 0
f x xlim 1
x f x
1.260 f : R R
3 2xf x 3f x e 1 , x R
) 2xf(x) e 1 , x R
) f
) x 0
1lim xfx
1.261 2
3
1 1x , x 2 xf xx x, x
) 0 f
ox .
) f 0
1.262 f : R R ,
2 2f(x) 2f(x) x 0 , x R
) f(x) 1 x
) f 0
) x 0
1lim xfx
1.263
1x
1x
2 2 , x 0f x2 1 , x 0
) :
x lim f x
, x -lim f x
, x 0lim f x
,
x 0lim f x
) f
;
1.264 f 0 f(x y) f(x) f(y)
x, y R R
1.265 f 1 f(xy) f(x) f(y) x, y 0, .
0,
1.266 f :
f x y f x f y 1 , x, y R .
f R ,
R .
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23
. 2016-2017
1.267
2x 1 x
x 2x
1 ,2 2
1.268 22 2 0
x x 1 x - 1
, , 0
, 1 2 , 1,1
2 2
21 2
-1 1
1.269 3 2x x 0 ,
, R , 0 , 1 0 .
1,1
1.270 ' f : , R , ,
2f() 2f() .
ox [,] , 2
o of(x ) x .
1.271 f : [0,] R ,
f(0) f() .
ox 0, , o of(x ) f x2
.
1.272 f R x R f(x) f(x 2) 0
:
A) f
B) R f() f( 2)
1.273 xf(x) 20 x e , g(x) 21 (x x) .
(,) 21y 20x ,
(,) 0,0 . fC , gC .
ox 0,1
1.274 f : R R
R f 4 f 4 0
f x 0 *x R .
f x f x 0 *x R f 0
1.275 f
1, 2 f 1 f 2 ,
ox 1,2 o3f( 1) 4f(2) 7f(x )
1.276 E
f : , R , f f , , .
ox (,) ,
07f x f 2f 4f
1.277 , 0 ,
x x ( )
.
1.278 f R f x f 2 x 0 x ,
f x 0
R .
1.279 f
1,2 f 2 6 , f 1 f 2 8 ,
, ox 1, 2
2o o of x x x .
1.280 f : 0, 0,
0
f f f 1
. ox
2016o of x x
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24 -
http://users.sch.gr/mipapagr
1.281 ' f : 0,1 0,1
. ox 0,1
o of x x (S. Banach)
1.282 f ,
R f x f xe 4x 4e 0 x R
f 0 ln 2
1.283 1 2 1994 , ,..., 0,1 .
, ox 0,1
o 1 o 2 o 1994x x ..... x 997.
1.284 f : R R
2f x 2f x x 1 , x R
1.285 * f : R R
2f x 9 x R f 0 3 .
f x 3 x R
1.286 N
f x 4 x 2 x
f x 0
1.287 f
224x 9 f(x) 36
x 3, 3 . f 0 2
1.288
) 21 xf x , 0 x 1
x
) 3f x x x , x 0, /2
1.289 f : R R : f 1 f 2 f 3 f 4 .
f ;
1.290 ,, R
0,2
2 2 2 3 2
1.291 f x x
g x 2x
0,4
.
1.292 f,g : 0,1 0,1
f g g f x 0,1 .
f
0,1 . () ox 0,1
o of x x o og x x
1.293
f 2 2f x 4x 5x x , 0 x 2
1.294 f : R Z
f 1 2 , f x 2 , x R .
1.295 f (0, + )
x 0lim f x R
x lim f x R
,
x 0
ox 1o of x e ln x 1 .
1.296 f R f f(x) x x R .
R f
1.297 f : R R f 10 9
x R f x f f x 1 .
f 5
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25
. 2016-2017
1.298 ) ln x 2 x 1 0
) 3 2
2
x 2 1 x 1 x 1f x
x x ln 1 x 1
R . f
R , R
1.299 2g x x xx 3 2x x x , g(x) 0
f x g(x) g(x) 0
R f
1.300 f : R R , 2 2 2f x x x , x R .
) f 0
) f R f f 0 0 .
1.301 f 1, 4 : f x 0 x 1, 4 , f 1 0
f 1 f 2 f 3 f 4 :
) f x 0 x 1, 4 ,
) 2g x f x f 1 f 2 1,2 .
) f .
1.302 - g(x) x5
, . 4 f g f()5
f
R , x [,] f(x ) f(g(x ))
1.303 f,g : 0, R g 0 1 x f(x) g(x) x 3f(x) g(x)
) f 0
) f x 0 x 0, 4 : ) x 2 f x xf x 2 x x 2
0, 2 .
) g x 0 x 0, 4 .
1.304 f : (0,1) R
x 0
f(x) 3lim 3x
22(x 1) (x 1)f(x) x 1 x (0,1)
) h(x) f(x) ln x 3
) f(x) 3g(x) e y x
0x (0,1)
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26 -
http://users.sch.gr/mipapagr
1.305 A) f 1 1 . ,, ,
f() f() f() f() f() f()
B) f 1 1 o ,
.
1.306 f , R 3 2 3 2f x 4f x 6f x x 2x 6x 1
x R . f x 0 0,1
1.307 f R , f f(x) x 4 2f x x R .
:
. f 1 1
. A f
. ox R o of x x
. f 1 1
1.308 - - 6 . 6 6 .
6 , 6 , ,
.
1.309 f , f x 0 , x , . 1 2 3 vx , x ,x , ..., x ,
1 2 , , 1 2 3 v
1f x f x f x ... f x
f v
2 1 2 3 vf f x f x f x ... f x
1.310 f : IR IR 2f x f f(x) 4 x IR f 2 1
A) f 0x 1 , x 11lim f(x) 3
x 1
B) f 1, 2
1.311 f f x ln x ln x 1
) f 1f
) xg x f x e
) xf x e 1 0
-
27
. 2016-2017
1.312 f : 0, IR 3 3f x xf x x 0 , x 0,
.
) 3x f x 0
)
2
f x2f x 1 , x 0x x
g x 1 , x 0f x
, x 0x
1.313 f, : , .
, .
1.314 f R R ,
3f x 3f x x x R .
) f 1 1 .
) f . ) 2
xf x
f x 3
x R
) f
) x 0
f x 1limx 3
1.315 ** f , f f . f
0,1 f 0 f 1 0 .
) f 12
.
) N f 1
, 1, 2,3...
1.316 f g :
f R f x 0 , x R , f 0 1 , f 2009 2009
f f(x) f f(x) 1f f(x) 2 f f(x) 1
, x R .
g 2g x f x f 1 f 2 , x R . ' :
) f x 0 , x R ) R f 2
) f 1 f 2 f 3 f 4 ) g 1, 2
) f g .
63 x2 x 4 2x 8 xf(x) x6 x 4
f(0) (0,1] 6f 6
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28 -
http://users.sch.gr/mipapagr
1.317 f,g : R R 2g x 7x 5x 2 , : 32 f x + 3f x = x + 5 ,
x R f R
1. , g , f
1f
2 f x x , y y
3. f 3x < x (_3) 1 xf x > g
4 x R 1x x g x h = xf
, , R {0} 2 2 2
4 4 4 2 2 2
1 1 1
+
1.318 1 : x
1 - x = 12
.
f R
f x
1 - f x = x + 12
x R
2 f 0 = 0
3 f R
4 : f x2 f x + 6 > 1 f xf 2 16f x - 3 > f 16 5 2 2x f x x f x
1.319 f : R R g :R R
x R f(f(x)) 2g(x) x
1 g R .
2 h(x) f(x) g(x)
0x R 0 0f(x ) x
3 fC gC .
4 0 0 0f(f(x x 2)) x x 2f(x x 2) 2
5 0 0f(f(lnx x 1)) ln x 1 x
1.320 f R f R R . f
fC A 1,5 B 3, 1 ,
1 f
2 1 1 2f 2 f (x x 3) 3
3 : 1 1 x x 1f 2 f (5 4 2 ) 3
4 2f x 5 4f x
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29
. 2016-2017
1.321 f : 0, R 0, , R
f xf e ln x x 0, . x ln x f x t x ln x x 0,
1) t 1 1
2) f x ln x
3) 2
2x x 1ln 1 xx 2
4) x e2 2
x e 2x ef f e 1 f xe f 2e2
1.322 f R , f x y f x f y
x,y R
1 f 0 f x f x 0 x R
2 f x y f x f y x,y R .
f x 0
3 f 1 1 .
4 1 1 1f (x y) f (x) f (y) x, y f(R)
5 2f 10 x f x f x 1
1.323 , f : R R ,
: f 2 , f 2 f(x) 2004 x R .
1 2x f x f 0, .
2 f , :
) ( , ) f .
) fC f y 2x
ox ,
3 2xxf(x) 4xlim
x 1
4 h : R R f(x) h(x) 2004x , x R .
f x 0 1 2, .
h x 0 1 2,
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http://users.sch.gr/mipapagr
1.324 f : R R xe 1 xe
xe 1
e ef(x)e e
1 f R f
2 f
3 1 xfe ( 1)e
, 0
4 3 2 4f x f x f x f x 0,
5 2 1
1 11
2 11
e elim fe e
1.325 f 0,1
2 2f (0) f (1) 13 6f(0) 4f(1) .
1 N f .
2 N 1x 2x (0,1)
f y 3x 1x
21 1 112f(x ) 3f 4f 5fe 2
3 1f(f (lnx 4) 1) 3
1.326 f : R R : f(1 ln x) 1 x lnx , x 0
1 : 1 xf(x) x e x R .
2 o f
3 xxe e 0,
4 xlim f(x)
, xlim f(x)
.
5 2 3f x f x f x 0,
1.327 f : R R 3f x f x 2x 0 x R
) f R
B) f R
) , R 2 2f f 2 2 5 0
) 3f x 3x 0
E) N 3 22x f x f x 1 0
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31
. 2016-2017
1.328 f : R R 3 2 3f (x) xf (x) 2 x x R ,
x 0
f(x)limx
x x
f(x) f(x)lim limx x
, , R 0 .
1 ,
2 g(x) f(1 x) 1 , x R x 1limg(x) 1
x 1
xg(x) 2 3 x g(x)lim
x 1
3 x
f(x) x 2007limf( x) x 2007
1.329 *f : R R 1 92f(x) f 6xx x
x R 0
1) 4f(x) xx
.
2) f .
3) 2x 1|f(x)| 3lim2x x 1
.
4) x 2
5lim f(x)x 2
.
5) fC 0,0
1.330 2f(x) ln x x 1 ln (1 )x , x 0 0 . 1 0 ,
xlim f(x)
A 0 :
2 f .
3 f
4 f
B5 2 7ln2 f x f x f x 0,
1.331 f : 1, R : 2f x 1 2ln x f x
x 1 3
2x
f 1 x x 1lim
x x 1
.
1 f 1 0
2 2f x lnx 1 ln x , x 1,
3 xlim f x
2x 0lim ln x 1 ln x 4 1, f
5 , 1, f f 3 4 2
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32 -
http://users.sch.gr/mipapagr
1.332 f : R R , x 0
f(x) 2lim 0x
1 f(0) 2
2 : 2
2x 0
x f(x)limx
f : 2 x 2xf (x) e f(x) e 1, x R
3 : x xf(x) e e , x R
4 xlim f(x)
, xlim f(x)
x
1f 2xlim
2 x
.
5 f ox 0
-
2016 - 2017
M
333
-
:
: 16.08
M.Ed. 2016
: http:users.sch.gr/mipapagr
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33
. 2016-2017
2 - 2.01
2f(x) x -5x 6 2
2.02
0 1xxe x 0f(x)
1 x x 0
2.03 x x 2
f(x) x 2 2 x 2x 2
, R f 2
2.04 x 3
f(x)lim 7x 3
f
0x 3 , f 3
2.05 f 0.
x f(x) x x x , x 0 .
f ox 0
2.06 f : R R 2 2f(x) x (x 1) x R ,
f ox 1
2.07 f,g : R R
R f g 3 .
:
) x
f(x) f()limx
)
2 2
x
(f(x)) (f())limx
) 2 2
x
f(x) x f()limx
)
x
g()f(x) f()g(x)lim
x
2.08 f
ox of(x ) 3 , of (x ) 2 . B x x
2f(x)-6limx-x
2.09 f : R R 0 1 f(0) f(1) .
1f(2x) x2g(x)1f(2x-1) x2
12
f (0) f (1)
2.10 f : R R ox 0
x 0
f(2x) f(x)lim 3x
. f 0 3
2.11 f : R R 1 f (1) 2 .
x
xlim (x 1) f(1) f 2x 1
2.12 f : R R ox R .
f(x) x xg x
f (x )(x-x ) f(x ) x x
ox
2.13 f : R R 0
f x y f x f y xy x, y R ,
f R .
2.14 f
x 0
f x 2lim 3
x
:
A) f
2 f (2) 3
B) :
i) 2
2x 2
f (x) f(x)limx 4
ii)
x
2x 1lim x fx 2
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34
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2.15 f : R R , 0 .
2 2
x 0
f (3x) f (2x)lim 2f(0)f (0)x
2.16 f 0x 1 f (1) :
f xy xf y yf x x, y 0, .
000
f(x )f x +x
00 x 1
2.17 ** f : R (0, )
3 4f (x) 2x f(x) 8 , x R
f
0x 0 f (0) 0
2.18 f R x R ,
h 0
f(x 3h) f(x 2h)lim 5f xh
-
2.19
) xef(x)
1 x
) 2
1f xx 4
) ln xg(x) xxx 1
) xln xf xe
). x xg(x)1 x
) ln xg(x)x 2
) 2 xf x1 x
) 2xf(x)
ln x
) x2x 1h(x)
e
) 1 xf(x)1 x
2.20 1 h
h 0
e elimh
,
x 1
x 0
e elimx
, x 0
x 12lim
x
2.21 P
2P x P x x R .
2.22 f : R R ox e .
x
x 1
f(e ) xf(e)lim ef (e) f(e)x 1
2.23 h
h 0
e 1limh
2.24
A) x
x 0
e 1lim 1x
B) 5 5
x 2
x 2lim 80x 2
2.25 g
R , g e 1 g e 2 .
2
2 xf x x g xln x
f e
2.26 f :
yxf x y e f y e f x xy
x, y R :
A) f 0 B) f 0 0
) R
oxo o of x f x f 0 e x , ox R .
) f 0
R
oxo o of x f x f 0 e x ox R
2.27 f : R R 0x , 0 ,
:
) x
f(x)ln x f()ln f()lim f ()ln x
) 2x f(x) xf() f()lim f ()
x x
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35
. 2016-2017
2.28 : 2 2f(x) x 3x ,
2 3f(x) (4x 1)
3 2f x ln x 3x ln 3
3f x ln 2x 2
x xf x 2 3 t , t R
4 32 3 2f x x 3 x 5 y , y R
2.29 :
) f x ln x , x 1
) x xf x log 2 3
) 4 32 3f x x 3 2x 5
2.30 :
) 2 1x x 0
f x x0 x 0
) 2f x x x 3 2
) log xf x x , x 0
) xf x x , x 0,2
) xf x 2
2.31 x 3f x e x x , x R .
) f
1f
) 1f 1fD ,
1 1f 1 2 .
2.32 f
0x 0 : 3 2 2f (x) x f(x) 2x x ,
x R f 0 .
2.33 ) f(x) c(x )(x )(x )
c,,, R x ,, :
f (x) 1 1 1f(x) x x x
) f 2 3 4 2
2
(x 5) (1 x )f(x)1 x
2.34 f R f x 0 x R .
) y f x
R .
) f 2 5 f 2 4
f 2 4
2.35 f
R 2 2f(x ) 3f 2x 1 5ln x 3x x 0 . f '(1) .
2.36 f : R R 0x , 0 ,
:
) x
f(x)ln x f()ln f()lim f ()ln x
) 2x f(x) xf() f()lim f ()
x x
2.37 f(x) x,x (0,)
) 1f
) 1f
,
1 21f (x) , x ( 1,1)
1 x
2.38 f f(x y) f(x)f(y) f(x) 0
x,y R . x 0
f(x) 1lim Rx
f R
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36
http://users.sch.gr/mipapagr
2.39 f 0 x R
f f(x) f x 2x . f 0 1 f 0 2
2.40 f,g
R x R 2f(x )g x e ,
f 1 0 , g 1 2g(1)f 1
2.41 P x
2P x P x
2.42
2 2x x , x 0
f x x0, x 0
f x ox 0
2.43 f R .
) f f
) f f
) f
:
) fC 0,0
) f x f x
) f 0 0
) f 2g(x) (x 1)f(x) 3x g (0) 3
2.44 f : 0, R
xf x x x e , x 0 . f 0,
2/ xf (x) x xx 2xe
2x 0f(x) 1lim
x
2.45 f : R R R .
0x R
fC 0x 1fC
0f x .
2.46 f
R . x 1
f (4 x)lim 5x 1
f 3 5
2.47 f R. :
) h 0
f (x 2h) f (x)lim 2f (x)h
, xR
) h 0
f (x h) f (x)lim f (x)h
, xR
) h 0
4f (x 2h) 6f (x h) 10f (x)lim 2f (x)h
x R
2.48 :
A) 2xy ln e 1 x y 1 y 1 y B) y ln x ln x
2x y xy y 0
2.49 f R x R
2f x xf x , f 1 0 .
2.50 :
) f x x , () f x x2
) xf x xe () xf x e x
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37
. 2016-2017
2.51 fC
0x 0 2
3
1x x 0xf(x)
x x 0
2.52 f
ox 1 2
x 1
f(x) xlim 7x 1
.
fC A 1,f 1
x 9y 5 0
2.53 3f(x) x .
fC
M( 2, 8)
2.54 f
: 2xln x f x x x x .
ox 1
fC
1,f(1) .
2.55 f : 0, R 1f xx
0 ,
Ox,Oy
ox
.
2.56 f gC , C
2f(x) x 2 21 1g(x) x8 2
y y .
2.57 2f x ln x x 3 ,
, R : 2x y 4 0
fC A 1,f 1 .
2.58 f f 2 x f 2 x 2x , x R .
2,f(2)
y x .
2.59 2x 2x x 2
f x x 2
x 1
,
,, R Cf
A(2, f(2))
2x y 1 0
2.60 2f x 4 x 2g x x 8x 20 .
fC gC .
2.61 0
2f x x 3x 1, f(1)
g xx
2.62 x xe ef
2(x)
x xe eg x( ) x2
.
2.63 f R f (x) 0
x R . gC g f(x)g(x)f (x)
x x ,
, x x
o45
2.64 **
4 2f x x 4x 3x .
f . (mathematica)
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38
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2.65 f : R R
: 2f x 2 x 3x 2 f x 3 2x 4 ,
x R .
1M ,02
fC
.
f
fC .
2.66 f x 2 ln x ,
x 0 , R .
fC M 1, f 1
R.
2.67 y 2x 0
y f(x) ,
ox 1 ,
gC 21g(x) f
x
1x 1
2.68 * 1f(x)x
xg(x) e ,
fC gC .
2.69
xg(x) e 2f(x) 2x ,
2.70 R f
xf(x) , y x .
2.71 f :
23f x 1 2f x 2 x 14x 5 , x R
) f .
) fC
1A 1,4
, .
2.72 ** f R , f ln x x ln x x , x 0 .
fC
ox 1 x x y y
2.73 ** ln (x)f xx
, x 0
)
fC x , f(x ) .
) A
x , f(x ) , ,
.
2.74
f : (0, ) R , 2f(x ) f(x) 3 ln x 4
)
f 1, f(1)
) : 2x 1x f(x) - 2lim
x - 1 .
2.75 x *f x e x, R
) B M fC
.
)
M R
2.76
21f(x) x 2x - 2(1 - ), R2
A)
.
B) fC
x x , y x .
) 0 ,
f
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39
. 2016-2017
2.77 x, y fC , f(x) x .
.
2.78 Oxy
xf x e , x 0 . M A, B
M Ox Oy . M
1 m /sec . ot 1,e , :
) OAM ) AB
) fC M , x x
2.79 A Ox Oy ,
. Oy
Ox
A v 10 km/min .
) ; ( )
) A O 0, 0
Oy .
) A O 0, 0
3 km ;
2.80 4m , . 1m . 2m
1m /sec . 6m , :
A) y t
6y t 3x(t)
, 2 x t 6
B)
2m .
2.81 *** O 2m/sec
().
( ),( ) , 2m
( ),( ) AB .
AB OAB
( www.mathematica.gr)
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40
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. Rolle ... 2.82 . Rolle
f x x 1 x x 0,1
2.83 2x x x 0f(x)
3 ( )x x 0
,, R
Rolle 1,1 1,1
f 0 .
2.84 f
3,2 2
3,2 2
.
0 3x ,2 2
o o of (x ) f(x )x .
2.85 f , , > 0
(, ) f f
.
, f f
2.86 f
, (, ).
,
2 3 3f f
3 f
2.87 f ,g
, , f() f()g() g()
g(x)g x 0 x (,) .
, f () f()g () g()
2.88 f
R f(x) 0 x R f(2) ef(1)
.
f (x) f(x)
1,2 .
2.89 f : [,] R , : 2 2 2 2f () f () .
, : f f
2.90 f 1,1 , 1,1 ,
42f 5 f(1) f( 1) .
2.91 f ,g
[,] (,)
f x 0 x [,]
ln f() ln f() g() g() .
(,) f () f() g () 0
2.92 f : R R .
f 1 f 0 f 0 f 0 0 . N
x 0,1 3f x 0 .
2.93 ) 3 2f x x x 3x 1
x R R 1 1 .
) . Rolle
f x 2g x e x x 3
2.94
xf(x) e 2x
x 3g(x) e x
y y
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2.95 N x x 11 2 3 0
2.96 xln 1 xe x
2.97 x x2 5 2 5x
2.98 x x5 x 4
2.99 ln x x 1 0
2.100 x xxe 1 e
2.101 : 2x x ln x 2
2.102 96 96 24x 3 x 1 16
2.103 5x 3x 0 R
2.104
2012 20112013x 2012 1 x 0 0,1 R
2.105 x 2e x x R
2.106 3 2x x x 0 2 , 0
R
2.107 8x 7x 6 R
2.108
xe x 1
xe x 1
2.109 N 3 2 ln x ln x ln x 0 , ,,, R
3 2 4 0
21,e
2.110 4 3 2x x 3x x 0
,,, R
, 2 8
----------------------------------------------------------------------------------------------
2.111 51 km .
0,6 .
85 km /h .
2.112 f 1, 5 f 1 2
f x 2 , x (1, 5) 10 f(5) 6
2.113 f 0,5
f 5 f 0 1 .
, 0,5 2f 3f 1
2.114 f
1, 4 x R f 4x 4f x
25f 1100
1 2 3 , , 1, 4 1 2 3f f f 12
2.115 f x log x .
1,20 19 loge1 log2
.
2.116
xx
x 0
2 2limx x .
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2.117 1 x 1 1lnx 1 x x
, x 0
2.118 2
2 1ln 1
, , R
2.119 , , R
2.120 x x1 x e 1 xe , 0 x 1
2.121 N :
) 1 1
x 1 xxe x 1 xe x 0 .
) e 2 ln e
2.122 N :
) x ln(x 1) xx 1
x>0
) x 1x e 1 (x 1)e x 1,2
2.123 x x 11 11 e 1 , x 0
x x
2.124 0 4
2
1 2 1 4 ( )4
2.125 f R R .
: f 1999 f 2002 f 2000 f 2001
***************************************************************
2.126 f R
fC , R
f 0 .
2.127 f
2, 2 2, 2
f( 2)= f(2)= 2 . f (x) 1 , x 2, 2
f(x) x , x 2, 2
2.128 f : R R .
f 1 f 0 f 0 f 0 0 . N
x 0,1 (3)f x 0 .
2.129 2 6 4 2f(x) x x x , *,, , R 2 23 5 .
.
2.130 R f f(ln) f(ln ) .
ln ln ln , ,, 0
2 e , 1 2 , R
1 2f ( ) f ( ) 0
2.131 f , , f f 0 .
ox ,
f x f x f x .
2.132 f : , R ,
, ,
f f 0 . :
) x , of x 0 ,
, f 0 ,
) x , of x 0 ,
, f 0 .
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43
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2.133 34x 2 x 1 0
0,1 R .
2.134 f , R f( 1) 1 , f(1) 1 .
) 1 21 1 1 2f f 2
) 1 21 1 1 2
1 1 2f '( ) f'( )
2.135 f
0, > 1 f(0) = 0 2f(x ) 2f(x), x [0,] .
1 2 , 0, 1 2f()f ( ) f ( )
2 - 1 .
2.136 f
2, 20
Rolle, :
) 1 2 , 2, 20
1 2 1 2f f 0 .
) 1 2 , (2,20) 1 2
1 23f ( )+ 2f ( ) = 0
) f (x) f(x) - f()
2, 20 .
) , , 2 20
2f 3f 4f 0
2.137 f : R R f 2 0 . R
,
f , f() , x x
P 2,0
2.138 f : 1, 4 R
f 1 2
f 4 8 .
fC .
2.139 f : , R
, , f 2 , f 2
) f x = 2x
, .
)
1 2 , , 1 2f f 4 .
2.140 04 3 2 ,
3 2f x x x x
0,1
2.141 N 3 2 ln x ln x ln x 0 , ,,, R
3 2 4 0
21,e .
2.142 f x x 1 ln 2x .
:
) 1 ,12
fC , f() x x
) x 2 2x2x e
1 ,12
2.143 f R .
f 0 f 10
f x2
.
ox 0,10
of x 0
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2.144 f : R R , :
f x 2f x f x 2f x , x R
f 0 f 0 f 0 1 . :
) xh x f x e
2 2g x f x f x 2 f x f x
) f .
2.145 f : R R
: f x f y x y 1
x,y R . f
2.146 f f 1 2x 7 12x ,
x R f 1 2
2.147 f 21f x
x , x R *
f 1 f 1 2
2.148 : ) f (x) f(x) x R
f(0) f (0) 1 xf(x) e , x R ,
) (x) (x) 5x x R ,
(0) 1 (0) 4 , x(x) e 5x , x R
2.149 f R
*R f 0 0 ,
. f
2,1 2,1
2.150 f R . f x 2x 1 f x , x R
c R 2x xf x ce
2.151 f , x R f (x) f(x) x x f(0) 1 .
2.152 f : 0, R
f 02
f x f x
x 0, f x x ,
R .
2.153 f : R R
: 2x 2 f x 2x 5x 2 , x R
f 3 7
2.154
f : 0, 0,
f (x) f(x) ln f(x) x 0 f (1) 0
2.155 N , , f R * x R *
f(x) xf (x) , f(1) 1 f( 1) 2 .
2.156 f f 0 2 ,
x xf(x) e f (x) e 0 , xR
2.157 M(0, 3)
24
4 1
2.158
f : 0, 0, , f (1) 0
f (x) f(x) ln f(x) x 0
2.159 f ,
R 2x[f (x) f(x)]e f(x) f (x)
x R f 0 1 .B f
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2.160 f : 0, R
f 02
f x f x
x 0, , f x x , R .
2.161 f R f x 0 , x R , f 1 9
M x, f(x)
4x f(x) , x R
2.162 f R , fC
O 0, 0 fC
O 0, 0
-2x y 3 0
2x 1 f (x) 4x f (x) 2f(x) 0, x R
2.163 f g
R f x 0 x R
f x g x f x g x
x R , f x g x
2.164 ) f : R R f x f x 0 , x R
f 0 f 0 0 . f
.
) g : R R
g x g x 0 x R g 0 0 ,
g 0 1 . g x x .
2.165 * f R f 0 0
f(x) 22xf x e 1 2x x 1 x R . f .
2.166 f : R R N .
2 f x f y x y , x,y R
f .
2.167 f , R ,
x, f(x)
x2xe f x 2A 1,e
f
2.168 * f : 0, + R
f xy f x f y x, y 0, +
f e e . f ox 1 .
f 0, +
f x e ln x , x 0, + .
2.169 f : R R ,
2f x y xy y f x
x,y R , f 1 1 , f 2 2 . f
R
2.170 f R
f 0 2 2xyf y x f y f x e ,
x,y R . :
) f x 0 x R f 0 1
) f R
) f 2x 2xf x e
2.171 f : 0, R ,
1 f 1 1
2 2f xy x f y y f x . x,y 0 .
) f
x 0
) 2f x x ln(x) x 0
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46
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- 2.172
) xf xln x
) f x x x, x [0,2)
2.173 N
) x
2
e ex 1 x 0f x
x ln x x 0
) 2 xf(x) ln(1 x ) e 1
2.174
2f x x x 1 20,3
2.175 N R ,
2 xf x x x 1 e , R .
2.176 f : R R f 0 0 f
,
f(x)g xx
, x 0 , .
2.177 O f g R f(0) g(0)
x R f (x)g(x) f(x)g (x)
g(x) 0 . : f(x) g(x)
x [0, ) f(x) g(x) x ( ,0] .
2.178 * f
[0,+ ) 5 3f(x) 2 f(x) 3f(x)
2 3x xx 1 ln x 1 x 1
2 6. f
.
-
2.179 ln(x 1)f xln x
, x 2
) f
) :
) 2ln e 1 ln e 1 . ) 2ln x 1 ln x 1 ln x , x 2
2.180
2ln(x 1) x x 6 0
2.181 2xln(x 1) x 0x 2
2.182 x 1e 2x e 0
2.183 x 2
x 1 x 1x 1 x
2.184 22ln(x) x , x 0,
2.185 f : R R
f 1 x f 1 x x R .
f x 0 , x R ,
f x 0
2.186 f : R R , f x 0
f x3 3 2f x ln f(x) e x x 2x 1
x R . 2f ln x f 1 x
2.187 xg x x g x x R ,
xg(x)x
x 0 .
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2.188 :
) 2f x x lnx ) xf x 2 , 0 x 2
) 2nxf xx
)
xef x2x
E) 2f x x 4 x
2.189 :
) x, x 0
f x 1 , x 0x
B) x 11 e , x 1f x
ln(1-x), x 1
2.190 22 xf x x e e .
2.191 , R
f x ln 2x x
x 1 2 ln 2 .
2.192
f : R R 2 2(f(x)) x 1 2xf(x) , x R .
f .
2.193 2f x x ln x .
fC f
.
2.194 R
3 2f x x 1 x 5 x 2
.
2.195 f : 0,1 R ,
f x 0
x 0,1 . 1 2x , x 0,1
1 2x x , 1 2f x f x 0
0,1 3f 0 .
2.196 R
2 xf x xe e .
2.197 f [,].
1 2x ,x (,) 1 2f(),f() f(x ), f(x ) ,
1 2 ( , ) f () 0 .
2.198 f,g : R R
:
f x x 1 g(x) xf x e e x x R .
fC A 0,1 ,
fC gC
ox 0
2.199 0 0(x , f(x )) , ox
f(x) x ln x x , R
R
2.200 f
0,3 f (x) 0 f(1) 1 , f(2) 1 .
2f(x)g(x)
1 f (x)
, 0 x 3 ,
g
2.201 f R . R
f() f () f () 0 f (x) 0 x ,
f (x) 0 ,
f (x) 0 f(x) 0 .
2.202 ** 2x 4x f x f x 0 , x 0, 4
f x 0 x 0, 4 .
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2.203 x 2 x 0,
2.204 f
R 3f x f x x , x 0, .
f x 0
0,
2.205
22 ln x x 1 0
2.206
28x x x 1 0
R
2.207 R
3 2x x 4x 0
2.208 0 x 22e 2 2x x R
2.209 ** 2f x x 1 ln(x) ,
: ) 2f ln(x 1) f 6 x x 0
) 17 3 2008f x f x f x f x
2.210
x 1 ln x x 1 ,
.
2.211 * , 0 ,
2 3x x
2.212 f : R R f x 0 x R .
xf e f x
2.213 f : R R
f x 2f x f 0 1 .
2xf x e x 0 .
2.214 f : R R f 0 0 f x f x 0
x R , xf x 0 x 0
2.215 f , 0,1 ,
0,1 f x f 1 f 0
x 0,1
2.216 f R , : f 0 1
2xe f x 1 0 , x R .
fC A(0,1)
2.217 x 2e x x 0 , R R
2.218 ln x x
, x 0 ,
2.219 , 0 ln x x 1
x x 2
x 0 , 1 .
2.220 x f x x , x>0 ,
>0 f x 0 , x 0 . e
f e, .
2.221 f R f 0 f 0 0 f x 2x x R .
1f 13
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2.222 f : R R ,
f x f x , x R
ox 0
f 0 0 . : x f x f x 0
2.223 xxf x 1 e
, R
)
f x 0 x R .
) 1 1e
xx 1g x 1 xe
.
2.224 f(x) x ln x , 0
)
x ln x x 0
)
f .
2.225 f R f 0 2 , f 0 0 ,
f x x 0 x R ,
3xf x 2 1
12
x R
2.226 f R x R
: 2 2f (3x 1) 4 4f(2x x 1) .
:
A (1, 4) : f () 0
B f
f (1) f (4)
f (x) 0
R
2.227
-
2.228 x x , x 0, 2 .
2.229 36x 6x x x 0
2.230 A)
x
vef(x) , N *x
B) v
x exev
, x (0, )
2.231 ) ee
) :
1821 1821 1
2.232 )
3 2f x 2x 2x x lnx , x 0
) ,, 0, 1 ,
3 3 3 2 2 22 3 2
2.233 f
2 xf x ln 1 x e
2.234 ** 2 xf(x) ln(1 x ) e 1
R 2f ln x f 1 x
2.235 A(1,1)
xf(x) e
2.236 0 , 0
ln ln ln2
2.237 x x 1x e , x 1,
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- -
2.238 .
) 2 8h(x) xx
) 5 3g(x) 3x 5x
). 2g(x) ln x x 1 ) xf(x) xe
2.239
xx ln e x , x IR f
2.240 N
2 2g x ln x 2x ln x x 3
2.241
5 4 3 2f x x 5x 10x x , x R , , R
, 2 .
2.242 f : 0, R
f x x xf xx f(x)
x 0 . N f
0, .
2.243
f x x ln x x , R ,
A 1,3
) 4 1 :
) fC
4 x ln x x 3 , x 1 .
2.244 f : R R
f(x)2(x x 1)f (x) xe 0 xR
fC
.
2.245 f R .
f ox
.
2.246 N IR
4 3 2 2f x 2x 4x 3 2 4 5 x x 1 x R , .
2.247 f : 0,1 R
2f x x 4 f x x 0 x 0,1 .
fC
2.248 f xf (x) 2x 0 , x R .
A(0, f(0))
fC
2.249 f R f R f x 0 ,
x R , f 1 0
g x f x f 2 x , x R .
g , g
gC
2.250 f : [0, ) R
f(0) 0 .
f(x)g(x)x
(0, ) .
2.251 x 2 22f(x) 2e x , 0.
f , (0, )
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2.252 f R
f(x) 2 xf(x) e 1 x x e x R .
)
) .
2.253 ) f .
1 2x ,x
1 21 2 f x f xx xf2 2
(Jensen)
B) :
2e 1 e 1 e 1
R
) ln ln ln2
, f,
2.254 x 0 , y 0 , 1 x y 1 ,
11 1 5x yx y 2
2.255 f R f x f x
f 0 f 0 0 . f
R
2.256 f R
f
, x R
3x3f x 4f4
2.257 f
1, f f 1 0
f x x 1 f x x 1, .
2.258 f
1,
f 1 1
f x x 1 f x 1 0 x 1, .
2.259 f : R R f x 0 f x 0
x R
2.260 f : R R x 1
x 1ef(x)
e 1
x R .
)
, .
) x 1
f(ln x) f (x 1) f(x 1) f (ln x)
2.261 f
1,
f 1 0 f x x 1 f x
x 1, .
2.262 ,, R ,
2.263 f
1,
f 1 1
f x x 1 f x 1 0 x 1, .
2.264 x 2f x e x x 1
) fC
.
) x 2e x x 1 .
) xe 1 x 1 x , x R
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52
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DE L HOSPITAL
2.265 )
xlim (x ln x)
) x 1lim lnx ln(lnx)
) 1x
xlim x e 1
)
1x
xlim x
2.266 :
) xx 0
x xxlimx e 1 x
)
x
x 1lim xlnx 1
) x
x 3limx 2
) x
xx
e 2x 1lim4e x 3
2.267
x ln x , 0 x 1
f x 1 x-1 , x=1
f 1 0, 5 .
2.268 N o 2x 0
1 1lim x x x
2.269 N 1x
x 0
elimx
1x 0
x
xlim
e
2.270 3 2
6 4x 0
1 x ln xlim
x ln x
2.271
x
xx
e 2x 1lim4e x 3
2.272
x x
x xx
3x ln x e elim
x ln x e e
2.273 N o x
ln x ln xlim
2 ln x
ln x
2.274 x 0
2x xlim1 x
2.275
4 x
2x 0
x e x 1lim 18
sinx x
2.276 x 2
22e 2x 2 xf x
x
f x
f xx 0
elime
2.277 f : R R
x xxf x e f x x e
x R . f 0 .
2.278 f : R R , , 1 x f x ln 1 x x
x 1 . f 0 . f
2 R 3f 02
f 0 f 0 0 . : x 0
f(x) f( x)lim 31 x
2.279 f : R R
R . x R
2h 0
f(x 4h) 2f(x 2h) f(x)lim 24x 8h
fC M 1, f(1)
y 5x 8 , f
2.280 1x
xln x x , x 0f x 1 , x=0
e ln( x) , x 0
, R f ox 0
2.281
,, x x
2x 0
e e lim 1x
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2.282 N
x
3eh(x)x
, ln xf(x)x 1
, 21
xk x xe
x x x ln e 1
2.283 f,g : 0, R
g x f x x ln x 1 ln x x 0 .
y x 3 fC
, gC .
2.284 N y 2x 2 ln 2
xf x 2 ln e 1 2 ln 2
2.285 f : R R g
xg x xf e . y 2x 1 fC 0 , gC
.
2.286 f : R R ,
x
1lim f x 1x
2
x
xf x xlim 2
ln x x
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2.290 f(x) xln x, x 0
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2.294 f f x xln x x 3
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2.302 3 2f x x x 12x , , R ,
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2.304 f(x) ln x x
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2.305 f x f(x) 2ln x f(x)f x e e x 0,
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2.306 f(x) x ln x , R
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2.308 * f , R , f xf x e x 1 , x R
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2.309 * g : 0, R 0, , g 1 2 , g 1 8 ,
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2.311 f R f x 0 x R .
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2.315 f , , ) f , f
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2.316 * f , ,
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2.317 f 1,e f 1 2 , f e e 1
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2.324 f 2x ln x , x> 0f x , x=0
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2.326 f : 2x f x 3 x x , x
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2.328 f xx 1f(x)
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f (x)
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f 1f x
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2.331 f,g : 0, R
y x 1 0x 0 .
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f 0 g 0 f 0 g 0 1
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x
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x
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3 3.01 f : R R f 2 e 2
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f
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R .
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f x 0
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0
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4
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10
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3.16 f R . ,, , R :
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3.17 1
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3
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3.19 f R . ,, , R :
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3.20 f R . ,, , R :
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3
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1 3
0 2f x g t dt dx
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x
F(x) f(t)dt
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f x dx F x dx F F :
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