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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    M

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    3 3

    x x

    3

    x

    2

    x

    lim f x f x 2 lim f x f x 2 0

    lim f x 1 f x 1 0

    lim f x 1 f x f x 1 f x 1 0

    lim

    2x

    f x 1 f x f x 2 0

    3 : (2/10/2016)

    )

    )

    )

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    x :

    22

    2 2

    2

    2

    2

    2

    f x 1 f x f x 2f x 1 f x f x 2f x 1

    f x f x 2 f x f x 2

    f x 1 f x f x 2

    1 7f x

    2 4

    f x 1 f x f x 2

    7

    4

    4 f x 1 f x f x 2

    7

    2

    2 2

    4 f x 1 f x f x 2f x 1

    7

    4 f x 1 f x f x 2 4 f x 1 f x f x 2f x 1

    7 7

    x

    24 f x 1 f x f x 2 4 00

    7 7lim

    x lim f x 1 0

    ,

    x limf x 1

    ) 0

    x 0

    f 0 limf x 1

    :

    f x

    x 0lim e f x e 1 e e 1 h h 1 1

    xh x e x R 1 1 R

    1 f 0 1

    )

    g x f x 2x 1 x R

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    g 0 f 0 1 0

    H g R g 0 f 0 2 2

    :

    f x 2x 1 f x 2x 1 0 g x g 0 x R

    H g Fermat

    2

    g 0 0 f 0 2

    ) F f R

    x R :

    2f x f x f x 2f x 2f x2

    h1 12

    2

    F x F x F x F x2

    F x F x

    F x F x

    e 2f x 2f x e f x f x f x

    h f x f x f x h 2f x 2f x

    f x f x f x 2f x 2f x

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

    f x e 2f x e 0

    f x e 2f x e c

    x 0 c 0 x R :

    F x F x 2x

    2x

    1

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

    f x e c

    x 0 1c 1 2xf x e x R

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    :

    x lim f x 1 0

    :

    2

    2

    3

    2

    2f x f x f x 1 f x f x 2f x 1

    f

    4f x 1 f x f x 2

    71 7f x

    2

    x 2

    4

    f

    2

    x

    3 34 4

    f x f x 2 f x 1 f x f x 27 7

    3 3

    x x

    4 4 4lim f x f x 2 lim f x f x 2 0 0

    7 7 7

    :

    x lim f x 1 0

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    f 0 2

    f x 2x 1,x

    f 0 1

    f 0

    R

    x 0

    x 0

    f x 1 f x f 02,x 0 2,x 0

    x xf x 2x 1,x f x 1 2x,xf x 1 f x f 0

    2,x 0 2,x 0x x

    f x f 02 f 0 2x f 0 2

    f x f 0 f 0 22

    x

    lim

    lim

    R R

    . -

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    :

    3x lim f (x) f(x) 2 0

    :

    3 3 2f () f(x) 2 f (x) 1 f(x) 1 (f(x) 1) f (x) f(x) 1 f(x) 1

    =

    32

    2

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

    f (x) f(x) 2

    2f (x) f(x) 2 > 0 < 0.

    3

    3

    2

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

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

    2 2

    2

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

    f (x) f(x) 2.

    (1)

    3 3f (x) f(x) 2 f(x) 1 f (x) f(x) 2

    3 31 f (x) f(x) 2 f(x) 1 f (x) f(x) 2

    3 3x x lim 1 f (x) f(x) 2 lim 1 f (x) f(x) 2 1

    x limf(x) 1 .

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    I :

    32 3

    2

    x x 2(x 1)(x x 2) x x 2 x 1

    x x 2

    f(x) 1=

    3 3

    2 2

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

    f (x) f(x) 2 f (x) f(x) 2 .

    33

    2

    f (x) f(x) 2f (x) f(x) 2

    f (x) f(x) 2.

    3f(x) 1 f (x) f(x) 2

    - 3f (x) f(x) 2 < f(x) 1< 3f (x) f(x) 2

    3 3x x lim f (x) f(x) 2 0 lim f (x) f(x) 2 0 ,

    x x lim(f(x) 1) 0 limf(x) 1

    )

    xg(x) e x xg'(x) e 1> 0 x R

    g R 1-1.

    f R ,

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

    g 1-1.

    ) f(x) 2x 1 (x) f(x) 2x 1 x=1,

    FERMAT (0) 0 f '(0) 2 '(x) f '(x) 2 .

    ) :

    2f ''(x) f(x)f '(x) 2 2f '(x) 2f (x)e 2f (x) 2f '(x) e f(x)f '(x) f ''(x) 2g(f ''(x) f(x)f '(x)) g(2f '(x) 2f (x))

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    g 1-1 :

    22f (x) 2f '(x) f(x)f '(x) f ''(x) 22f (x) f(x)f '(x) 2f '(x) f ''(x)

    f(x)(2f(x) f '(x)) = 2f '(x) f ''(x)

    f(x)(2f(x) f '(x)) - ( 2f '(x) f ''(x) ) = 0 (1)

    F(x) f , (1) :

    F'(x)(2f(x) f '(x)) ( 2f '(x) f ''(x) )= 0

    - F(X)e F'(x)(2f(x) f '(x)) F(x)e ( 2f '(x) f ''(x) ) = 0

    F(x)e (2f(x) f '(x) =0 F(x)e (2f(x) f '(x) = c

    x = 0 c = 0 .

    2f(x)-f(x) = 0 2x 2x2e f(x) e f '(x) 0 2xe f(x) =0 2x 1e f(x) c

    f(0) 1 1

    c 1 , 2xf(x) e .

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    3 3 2x x x lim f (x) f(x) 2 lim f (x) f(x) 2 0 lim f(x) 1 f (x) f(x) 2 0 ,

    2g(x) f(x) 1 f (x) f(x) 2

    x limg(x) 0 (1)

    2

    2 2 1 1 1 1 7 7f (x) f(x) 2 f (x) 2 f(x) 2 f(x) 12 4 4 2 4 4

    ,

    g(x) 02

    2 2

    g(x)1f (x) f(x) 2 1 0 1 g(x)

    f (x) f(x) 2 f (x) f(x) 2

    (1)

    2 2

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

    f (x) f(x) 2 f (x) f(x) 2 ,

    x x lim g(x) lim g(x) 0

    x x lim f(x) 1 0 limf(x) 1

    ) f ,

    x 0limf(x) f(0)

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

    g x xg(x) e x, x , g (x) e 1 0 , gr 1 "1 1" , (1)

    "1 1"

    g(1) g f(0) 1 f(0)

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

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

    h(x) h(0) . x 0 h (x) f (x) 2

    .Fermat

    h (0) 0 f (0) 2 0 f (0) 2

    )

    2f (x) f (x)f(x) 2 2f (x) 2f (x)e 2f (x) 2f (x) e f(x)f (x) f (x) ,

    2)

    f (x) f (x)f(x) 2f (x) 2f (x) 2 2

    g "1 1"

    e f (x) f(x)f (x) e 2f (x) 2f (x) g f (x) f(x)f (x) g 2f (x) 2f (x)

    f

    2

    2

    (x) f(x)f (x) 2f (x) 2f (x)

    f (x) 2f (x) f(x)f (x) 2f (x)

    f (x) 2f (x) f(x) f (x) 2f(x)

    A(x) f (x) 2f(x) , A (x) f (x) 2f (x)

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

    F f , . F (x) f(x) , ( f ) ,

    F( x)e 0F(x) F(x) F(x) F(x)A (x) F (x)A(x) 0 A (x)e F (x)e A(x) 0 A(x)e 0 A(x)e c

    x 0

    F(0) F(0)A(0)e c f (0) 2f(0) e c 0 c ,

    F( x) 2 xe 0 e 0F(x) 2x 2x 2x

    2x 2x

    1 1

    A(x)e 0 A(x) 0 f (x) 2f(x) 0 e f (x) 2e f(x) 0 e f(x) 0

    e f(x) c f(x) c e

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

    2xf(x) e ,x R

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  • ___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

    ) f , :