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    lisari.blogspot.gr

    http://lisari.blogspot.grhttp://lisari.blogspot.gr

  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    1

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    2

    5

    (1-10-2015)

    10 2015

    f : :

    x

    e x , x 0f (x)

    f(0) x ln(x 1), x 0

    , , , :

    f (x) 1 , x .

    . : 1 1

    . :h 0

    f (x 2h) f (x h) 2f(x)lim f(x)

    h

    , x

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

    . , , :

    f () f () 10

    x 1 x 1

    1,1 .

    . :2 2 xf (x ) x e 2 .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    3

    1 ( )

    . f f f ox 0

    xx 0 x 0lim f(x) f(0) lim e x f(0) f(0) 1

    .

    f (x) 1 f (x) f (0) .

    f ox 0 .

    ox 0 .

    f ox 0 .

    Fermat, :

    +

    x

    x 0 x 0

    1 x ln x 1 1e x 1lim lim 0

    x x

    +

    x

    x 0 x 0

    ln x 1e 1lim lim 0

    x x

    1 1 0

    1 1 ,

    0xx 0

    x

    DLHx 0 x 0 x 0

    e 1e 1lim lim lim e 1

    x (x)

    0

    0

    DLHx 0 x 0 x 0

    ln(x 1)ln(x 1) 1lim lim lim 1

    x (x) x 1

    .

    .

    xe 1, x 0

    f (x) 0, x 0

    x, x 0

    x 1

    x x xx 0 x 0 e 1 e 1 1 e 0 .

    x 0 x

    0 1x 1

    f (0) 0 ,

    f (x) 1 , x .

    : h 0 h 0

    f (x 2h) f (x h) 2f (x) f (x 2h) f (x) f (x h) f (x)lim lim

    h h h

    +x 0 x 0

    f(x) f(0) f(x) f(0)f (0) 0 lim lim 0

    x x

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  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    4

    h 0 h 0

    f (x 2h) f (x) f (x h) f (x)lim lim 2f (x) f (x) f (x)

    h h

    ,

    2h H

    h 0 H 0 H 0

    f (x 2h) f (x) f (x H) f (x)lim lim f (x)

    h H

    H h

    h 0 H 0 H 0

    f (x h) f (x) f (x H) f (x)lim lim f (x)

    h H

    .

    f (x) f(x) , x

    f(x) 1 f (x) 1 .

    . :

    x

    2

    e , x 0

    1f (x), x 0

    x 1

    f (x) 0 x

    f ox 0 , +x

    x 0 x 0

    xlim e 1 lim 0

    x 1

    ,

    f .

    .. f x,0 x 0 ,

    1 x,0 1f(x) f(0) f(x) 1

    f ( )x x

    .

    f . x 0

    1 1

    f(x) 1x 0 f (x) f ( ) f (x) xf (x) f(x) 1

    x

    f(x) xf (x) 1 .

    .. f 0, x x 0 ,

    2 0, x 2f(x) f(0) f(x) 1

    f ( )x x

    .

    f . x 0

    2 2

    f(x) 10 x f (x) f ( ) f (x) xf (x) f(x) 1

    x

    f(x) xf (x) 1 .

    f f . ,0

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

    x 0 f(x) f(0) f(x) 1 .

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

    . g

    g(x) f () x 1 f () 1 x 1 ,

    x 1,1 , 1,1

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  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    5

    g( 1) 2 f () 1 0 , f() 1 g(1) 2 f () 0 ,

    .

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

    Bolzano g . 1,1

    g(x) 0, 1,1 ,

    : f () f () 1

    0x 1 x 1

    , 1,1 .

    . x 2x 0 , 2 2 2f(x ) 1 x ln x 1 . : 2 2 xf (x ) x e 2 2 21 x ln x 1 2 xx e 2

    2 xln x 1 e 1 0.

    m

    2 xm(x) ln x 1 e 1, x . x22x

    m (x) ex 1

    .

    :

    x 0 , m (x) 0 .

    x 0 , x x2

    2xe 1 e 0 m (x) 0

    x 1

    , 2

    2x1

    x 1

    ,

    x .

    m (x) 0 x , m

    1-1.

    :m 1-1

    m(x) 0 m(x) m(0) x 0 .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    6

    2 ( ) ( )

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

    x 0 f(x) > f(0) f(x) > 1

    x > 0

    nx x 1 x = 1

    x x + 1 > 0

    n(x 1) x 1 1 x n(x 1) 0 1 x n(x 1) 1 f (x) 1

    x + 1 = 1 x = 0

    x 0 : f(x) > 1 (1)

    x

    2

    e , x 0

    f (x) 1 , x 0

    (x 1)

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

    ( x 0 x 0im f (x) im f (x) f (0)

    )

    f .

    f :

    [x, 0] [0, x]

    (x, 0) (0, x)

    1(x, 0)

    2(0, x)

    1

    f (0) f (x) 1 f (x)f ( )

    0 x x

    2

    f (x) f (0) f (x) 1f ( )

    x 0 x

    f

    x < 1 < 0

    11 f (x)

    f (x) f ( ) f (x) xf (x) 1 f (x) f (x) xf (x) 1x

    0 < 2 < x

    2 2

    f (x) 1x f ( ) f (x) f (x) f (x) 1 xf (x) f (x) xf (x) 1

    x

    x 0 : f (x) xf (x) 1 (2)

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  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    7

    (1), (2) x 0 : 1 < f (x) xf (x) 1

    .

    2 2 xf (x ) x e 2 21 x 2 2n(x 1) x xe 2 x 2e n(x 1) 1 0

    x 2(x) e n(x 1) 1

    x

    2

    2x(x) e

    x 1

    x 2

    1 2x

    e x 1

    2 x

    x 2

    x 1 2xe

    e (x 1)

    2 x

    x 2

    x 1 2x 2xe 2x

    e (x 1)

    2 x

    x 2

    (x 1) 2x(e 1)0

    e (x 1)

    x < 0 ex < e0 ex < 1 ex 1 < 0 2x(ex 1) > 0

    x > 0 ex > e0 ex > 1 ex 1 > 0 2x(ex 1) > 0

    x 0 2x(ex 1) > 0

    x 1 2

    x 2

    (x 1)0

    e (x 1)

    2 2 xf (x ) x e 2 (x) = (0) x = 0 ( 1 1 )

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  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    8

    6

    (11-10-2015)

    18/10/2015

    f : R R

    xf (f (x)) ln(4 e ) (1) x R

    xg(x) f(x) ln(2 e ) (2).

    :

    . f 1-1

    . 0

    x R 0

    g(x ) 0

    . g(x) 0 0f (x )

    . f , R

    e

    f ( )2 e

    : -

    - -

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  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    9

    1 ( )

    . 1 2x ,x 1 2f (x ) f (x )

    1 2 1 2

    1 2

    x x x x

    1 2

    x x

    1 2

    f (f (x )) f (f (x )) ln(4 e ) ln(4 e ) 4 e 4 e

    e e x x

    f 1-1.

    . x R g(x)>0 g(x)

  • ___________________________________________________________________________ 2015 http://lisari.blogspot.gr

    10

    . x0 f(x0) g(x) 0 g

    ( ) g

    [x0 , f(x0)] [f(x0), x0] Rolle,

    g ( ) 0 . x

    x

    eg (x) f (x)

    2 e

    (4)

    (4) e e

    g ( ) 0 f ( ) 0 f ( )2 e 2 e

    2 ( ) ( )

    . g(x) 0 x . g

    , ,

    ,

    : g(x) 0 x g(x) 0 x .

    g(x) 0 x .: xf(x) ln 2 e , x , x f(x)

    f(x) x f(x) x f(x)f f(x) ln 2 e ln 4 e ln 2 e 4 e 2 e x f(x)2 e e xf(x) ln 2 e .

    , g(x) 0 x

    g(x) 0 ,

    ox , og(x ) 0 .

    . o of(x ) x

    o oe 2 e f(x ) x .

    g ,

    x

    x

    eg (x) f (x) , x

    2 e

    o og(x ) g f(x ) 0 ,

    Rolle g o ox ,f(x ) ,

    o ox ,f(x ) , g () 0

    ef () .

    2 e

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