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• 18 05 16

(edit) 19 : 00 .

lisari team

2016

4

• lisari team

http://lisari.blogspot.gr/2014/10/blog-post_13.html

4 : 18 05 2015 ( )

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http://lisari.blogspot.gr

http://lisari.blogspot.gr/2014/10/blog-post_13.htmlhttp://lisari.blogspot.gr/

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lisari tea

18 05 2016

mailto:[email protected]

• lisari team

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• http://lisari.blogspot.gr

18 05 2016

1

lisari team / 2015 16

: 18 2016

:

&

: ENNIA (9)

1. , .262, i

f (x) 0 0x (,x ) f 0x , f

0(,x ] .

0f (x) f (x ) , 0x (,x ] . (1)

f (x) 0 0x (x ,) f 0x , f

0[x ,) . :

0f (x) f (x ) , 0x [x ,) . (2)

y

O

f (x0)

f 0

a x0 x

, (1) (2), :

0f (x) f (x ) , x (,) ,

0f (x ) f (,) .

A2. , .141

f g :

x A f (x) g(x) .

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

2

f g f g .

A3. , .246-247

f :

[,]

(,)

, , (,) , : f () f ()

f ()

(,) f f

f

, , M(,f ()) fC M

.

A4. ()

()

()

()

()

(,f ())

x

y

M(,f ())

A(,f ())

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

3

B1. x R :

22

2xf x

x 1

f :

f 0, , ,0 () (0, 0).

2. x R :

2

32

2 1 3xf x

x 1

:

f 3

,3

3,

3

,

3 3,

3 3

3 3 3 3

A ,f ,B , f3 3 3 3

.

3. f

R . :

2 2

2 2x x x

x xlim f x lim lim 1

x 1 x

xlim f x 1

f

y = 1 .

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

4

4. f , f x f x x R xx

. f x 0 x R , x = 0. f xx (0, 0).

f :

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

5

1. :

ln x x 1 x 0 , x 1 .

x 2xe :

2 2 2 2x x 2 x x 2ln e e 1 x e 1 e x 1 0

2x 2e 1 x 0 x 0 .

2x 2e x 1 0 x 0 .

2. :

2 222 x 2 x 2f x e x 1 f x e x 1

2x 2e x 1 0 x R .

2x 2f x 0 e x 1 0 x 0

f R ,0 , 0, .

:

x < 0 f x 0 2x 2f x e x 1

x < 0 f x 0 2x 2f x e x 1

x > 0 f x 0 2x 2f x e x 1

x > 0 f x 0 2x 2f x e x 1

f(0) = 0 :

2x 2f x e x 1, x R

2x 2f x e x 1 , x R

2

2

x 2

x 2

e x 1 , x 0f x

e x 1 , x 0

2

2

x 2

x 2

e x 1 , x 0f x

e x 1 , x 0

3. x R :

2xf x 2x e 1

2 22 x xx 0 e 1 e 1 0 f x.

x R :

2 2 2 22 x x 2 x xf x 4x e 2e 2 4x e 2 e 1 0

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

6

f R 0x 0 , f x 0

x ,0 0, f R f R .

2 22 x xf x 2 2x e 1 e

x 0 f 0 0 , :

2x2 2 2 2 2

2 2

e2 x x 2 x x 2 x

2 x 2 x

f x 0 2x e 1 e 0 2x e 1 e 2x e 1

2x 1 e 2x 1 e 1

x 0 , 22x 1 1 2x 0e e 1 .

4. x 0 .

g x f x 3 f x , gD R

g x f x 3 f x , x R f R f R .

f

x 3 x f x 3 f x f x 3 f x 0 g x 0

0 x f (x) < 0 x .

, f (0) = 1 > 0, f (x) > 0 x f . .

http://lisari.blogspot.gr/

• http://lisari.blogspot.gr

18 05 2016

8

3.

x +

( ) f (x) 0

+

x + x +

f

lim f(x) = +f

f ( )

-2 x x 2

-1 x 1 f (x) f (x) f (x) - 2 x x 2

-1 x 1 -2 -2lim lim 0

f(x) f(x)