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Δη μ ήτρης Αντ . Μοσχόπουλος Καθηγητ'ς Μαθη μ ατικ,ν Πτυχιο2χος Αριστοτελε8ου Πανεπιστη μ 8ου Θεσσαλον8κης Μαθηματικά Γ΄ Λυκείου Διαφορικός Λογισμός Νέα Μουδανιά - Ιούνιος 2015 - 2η έκδοση Θεώρημα του Rolle 10 αναλυτικ( λυ μ )νες ασκ-σεις ΚΑΤΗΓΟΡΙΑ 37 Ασκήσεις με συμπέρασμα για την δεύτερη παράγωγο μιας συνάρτησης .

Κ37-Θεώρημα Rolle (5)

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Text of Κ37-Θεώρημα Rolle (5)

  • . ' ,

    2 8 8 8

    - 2015 - 2

    Rolle

    10( ) -

    )37((((((((

    .

  • ( 1 - 4)

    ' 2.

    .

    , .

    4 , 2010.

    ( 5 - 10)

    :

    37

    .

    . , .

    6. Rolle.

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  • 1. f :! ! , , x'x 1, 2, 3. , , (1,3) , f () = 0 .

    2. f :! ! , , f (0) =1 , f (1) = e f (2) = e

    2 . , , g(x) = f (x)e

    x + x 23x .

    ) , , C

    g

    (0,2) , Cg .

    ) , , (0,2) , f ()+ 2 = e .

    3. f :! ! , . f x'x x1 x2 , x1 < x2 . , , ! , f

    (3)() = 0 .

    4. f : [1,3] ! , , f (1) = 2 , f (2) = n(2e

    3) f (3) = n(3e4) . :

    ) g(x) = f (x) nx x - [1,3] .

    ) , , (1,3) , 2 f () = 1 .

    5. f [0,1] , f (0) = 0 . : ) g(x) = f (x)x f (0)[f (1) f (0)]x

    2 - Rolle [0,1] .

    ) , , (0,1) , f (1) f (0) =

    12 f () .

  • 6. f, [,] , f () = f () = f () = f () = 0 .

    : ) g(x) = e

    x [ f (x) f (x)] Rolle [,] .

    ) , , (,) , f () = f () .

    7. f, [,] - (,) ,

    f ()f ()

    =f ()f ()

    f (x) 0 , x ! .

    x0 (,) , f (x0) f (x0) = [ f (x0)]2 .

    8. , f, ! , f (1) = 1 , f (2) = 4 n2 , f (e) = e

    21 .

    (1,e) ,

    f () =12

    + 2 .

    9. f, ! , f (0) = 0 , f () = f () =

    2 .

    ( ,) , f () = 2 .

    10. f, ! , f (1) = f (1) = 1 , . (1,1) , f , ( , f ()) :y = 2x 3 .

  • 1. f :! ! , , x'x 1, 2, 3. , , (1,3) , f () = 0 .

    C

    f x'x 1, 2, 3, f (1) = f (2) = f (3) = 0 .

    f ! , f , f , :

    f , , ! .

    f , , ! .

    . f [1,2] , [2,3] .

    . f (1,2) , (2,3) .

    . f (1) = f (2) = f (3) = 0 . Rolle ,

    x1 (1,2) , f (x1) = 0 , x2 (2,3) , f (x2) = 0 .

    . f [x1 ,x2 ] .

    . f (x1 ,x2) .

    . f (x1) = f (x2) = 0 .

    Rolle, (x1 ,x2) , (1,3) ,

    f () = 0 .

    . , .

    6. Rolle.

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  • 2. f :! ! , , f (0) =1 , f (1) = e f (2) = e

    2 . , , g(x) = f (x)e

    x + x 23x .

    ) , , C

    g

    (0,2) , Cg .

    ) , , (0,2) , f ()+ 2 = e .

    ) , , f, ' - , f () = 0 .

    , x1 ,x2 (0,2) , g (x1) = g (x2) = 0 .

    f ! , f , f , :

    f , , ! .

    f , , ! .

    . g [0,1] , [1,2] .

    . g (0,1) , (1,2)

    . g(0) = f (0)e0 + 02 3 0 =11 g(0) =2 .

    IV. g(1) = f (1)e112 3 1 = ee +13 g(1) =2 .

    V. g(2) = f (2)e2 + 22 3 2 = e2 e2 + 46 g(2) =2 .

    Rolle ,

    x1 (1,2) , g (x1) = 0 , x2 (1,2) , g (x2) = 0 .

    ) g (x) = f (x)ex + 2x 3 .. g [x1 ,x2 ] .

    . g (x1 ,x2) .

    . g (x1) = g (x2) = 0 ().

    Rolle, (x1 ,x2) , (0,2) ,

    g () = 0 (1)

    g (x) = f (x)ex + 2 , (1)

    f ()e + 2 = 0 f ()+ 2 = e .

    . , .

    6. Rolle.

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  • 3. f :! ! , . f x'x x1 x2 , x1 < x2 . , , ! , f

    (3)() = 0 .

    C

    f x'x x1 ,x2 , f (x1) = f (x1) = f (x2) = f (x2) = 0 .

    f ! , f , f , f(3) , :

    f , , ! .

    f , , ! .

    f , , ! .

    . f [x1 ,x2 ] .

    . f (x1 ,x2) .

    . f (x1) = f (x2) = 0 .

    Rolle, x0 (x1 ,x2) , f (x0) = 0 .

    . f [x1 ,x0 ] , [x0 ,x2 ] .

    . f (x1 ,x0) , (x0 ,x2) .

    . f (x1) = f (x0) = f (x2) = 0 .

    Rolle ,

    1 (x1 ,x0) , f (1) = 0 , 2 (x0 ,x2) , f (2) = 0 .

    . f [1 ,2 ] .

    . f (1 ,2) .

    . f (1) = f (2) = 0 .

    Rolle, (1 ,2) , f(3)() = 0 .

    . , .

    6. Rolle.

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  • 4. f : [1,3] ! , , f (1) = 2 , f (2) = n(2e

    3) f (3) = n(3e4) . :

    ) g(x) = f (x) nx x - [1,3] .

    ) , , (1,3) , 2 f () = 1 .

    ) , , f, ' - , f () = 0 .

    , x1 ,x2 (1,3) , g (x1) = g (x2) = 0 .

    f [1,3] , f , f , :

    f , , [1,3] .

    f , , [1,3] .

    . g [1,2] , [2,3] .

    . g (1,2) , (2,3)

    . g(1) = f (1) n11 = 21 g(1) = 1 .

    IV. g(2) = f (2) n22 = n2e3 n22 = n2 + ne3 n22 = 32 g(2) = 1 .

    V. g(3) = f (3) n33 = n3e4 n33 = n3 + ne 4 3 = 43 g(3) = 1 .

    Rolle ,

    x1 (1,2) , g (x1) = 0 , x2 (2,3) , g (x2) = 0 .

    ) g (x) = f (x)

    1x1 .

    . g [x1 ,x2 ] .

    . g (x1 ,x2) .

    . g (x1) = g (x2) = 0 , ' ().

    Rolle, (x1 ,x2) , (1,3) ,

    g () = 0 (1)

    g (x) = f (x)+1x 2

    , (1)

    f ()+

    12

    = 0 2 f ()+1 = 0 2 f () =1 .

    . , .

    6. Rolle.

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  • 5. f [0,1] , f (0) = 0 . : ) g(x) = f (x)x f (0)[f (1) f (0)]x

    2 - Rolle [0,1] .

    ) , , (0,1) , f (1) f (0) =

    12 f () .

    ) f [0,1] , f , f , : f , , [0,1] .

    f , , [0,1] .

    . g [0,1] - .

    . g (0,1) - .

    . g(0) = f (0)0 f (0)[f (1) f (0)]02 = f (0) g(0) = 0 .

    IV. g(1) = f (1)1 f (0)[f (1) f (0)]12 = f (1) f (0) f (1)+ f (0) g(1) = 0 .

    g Rolle [0,1] .

    ) g Rolle [0,1] , x0 (0,1) , g (x0) = 0 .

    g (x) = f (x) f (0)2 [f (1) f (0)]x .

    g (0) = f (0) f (0)2 [f (1) f (0)]0 g (0) = 0 = g (x0) .

    . g [0,x0 ] .

    . g (0,x0) .

    . g (0) = g (x0) = 0 .

    ' Rolle, (0,x0) , (0,1) ,

    g () = 0 .

    g (x) = f (x)2 [f (1) f (0)] ,

    f ()2 [f (1) f (0)] = 0 2 [f (1) f (0)] = f () f (1) f (0) =

    12 f () .

    . , .

    6. Rolle.

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  • 6. f, [,] , f () = f () = f () = f () = 0 .

    : ) g(x) = e

    x [ f (x) f (x)] Rolle [,] .

    ) , , (,) , f () = f () .

    ) f [,] , f , f , : f , , [,] .

    f , , [,] .

    . g [,] - .

    . g (,) - .

    . g() = e [ f () f ()] = e 0 g() = 0 .

    IV. g() = e [ f () f ()] = e 0 g() = 0 .

    g Rolle [,] .

    ) g Rolle, (,) ,

    g () = 0 (1)

    g (x) = (ex ) [ f (x) f (x)]+ex [ f (x) f (x) ]

    g (x) = ex [ f (x) f (x)]+ex [ f (x) f (x)] = ex [ f (x) f (x)+ f (x) f (x)]

    g (x) = ex [ f (x) f (x)] .

    (1) e [ f () f ()] = 0

    e>0

    f () f () = 0 f () = f () .

    . , .

    6. Rolle.

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  • 7. f, [,] - (,) ,

    f ()f ()

    =f ()f ()

    f (x) 0 , x ! .

    x0 (,) , f (x0) f (x0) = [ f (x0)]2 .

    x0 = x

    f (x) f (x) = [ f (x)]2 f (x) f (x)[ f (x)]2 = 0 [ f (x) ] f (x) f (x) f (x) = 0

    f (x )0

    [ f (x) ] f (x) f (x) f (x)f 2(x)

    = 0 f (x)

    f (x)

    = 0 .

    g(x) =

    f (x)f (x)

    , x [,] , x0 (,) , g (x0) = 0 .

    f [,] (,) , f , f , :

    f (,) .

    f , , (,) .

    . g [,] .. g (,) .

    III. g() =

    f ()f ()

    , g() =f ()

    f ().

    f ()f ()

    =f ()f ()

    , ,

    f ()f ()

    =f ()

    f () g() = g() .

    Rolle, x0 (,) , g (x0) = 0 .

    . , .

    6. Rolle.

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  • 8. , f, ! , f (1) = 1 , f (2) = 4 n2 , f (e) = e

    21 .

    (1,e) ,

    f () =12

    + 2 .

    = x

    f (x) =

    1x 2

    + 2 f (x)1x 2

    2 = 0 f (x)+1x

    2x

    = 0 [f (x)+ nx x 2 ] = 0 .

    g(x) = f (x)+ nx x2 (1,e) ,

    g () = 0 .

    f ! , f , f , :

    f , , ! .

    f , , ! .

    . g [1,2] , [2,e] .

    . g (1,2) , (2,e) - .. g(1) = f (1)+ n11

    2 g(1) = 0 .

    IV. g(2) = f (2)+ n222 = 4 n2 + n2 4 g(2) = 0 .

    V. g(e) = f (e)+ nee2 = e21+1e2 g(e) = 0 .

    Rolle ,

    x1 (1,2) , g (x1) = 0 , x2 (2,e) , g (x2) = 0 .

    g (x) = f (x)+

    1x2x .

    . g [x1 ,x2 ] .

    . g (x1 ,x2) .

    . g (x1) = g (x2) = 0 .

    Rolle, (x1 ,x2) , (1,e) ,

    g () = 0 .

    . , .

    6. Rolle.

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  • 9. f, ! , f (0) = 0 , f () = f () =

    2 .

    ( ,) , f () = 2 .

    = x

    f (x) = 2 x f (x)2 + x = 0 [ f (x)2x x ] = 0 [f (x)x2 x ] = 0 .

    g(x) = f (x)x2 x ( ,) , -

    g () = 0 .

    f ! , f , f , :

    f , , ! .

    f , , ! .

    . g [ ,0] , [0,] .

    . g ( ,0) , (0,) - .. g() = f ()()

    2 () = 22 + g() = 0 .

    IV. g(0) = f (0)02 0 g(0) = 0 .

    V. g() = f ()2 = 22 g() = 0 .

    Rolle ,

    1 ( ,0) , g (1) = 0 , 2 (0,) , g (2) = 0 .

    g (x) = f (x)2x x .

    . g [1 ,2 ] .

    . g (1 ,2) .

    . g (1) = g (2) = 0 .

    Rolle, (1 ,2) , ( ,) , - g () = 0 .

    . , .

    6. Rolle.

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  • 10. f, ! , f (1) = f (1) = 1 , . (1,1) , f , ( , f ()) :y = 2x 3 .

    f f () (),

    f () =

    f () = 2 f ()2 = 0 (1)

    (1,1) , (1).

    = x (1)

    f (x)2 = 0 [ f (x)2x ] = 0 [f (x)x2 ] = 0 .

    g(x) = f (x)x2 (1,1) ,

    g () = 0 .

    f ! , f , f , :

    f , , ! .

    f , , ! .

    . g [1,0] , [0,1] .

    . g (1,0) , (0,1) - .. g(1) = f (1)(1)

    2 = 11 g(1) = 0 .

    IV. g(0) = f (0)02 g(0) = 0 .

    V. g(1) = f (1)12 = 11 g(1) = 0 .

    Rolle ,

    1 (1,0) , g (1) = 0 , 2 (0,1) , g (2) = 0 .

    g (x) = f (x)2x .

    . g [1 ,2 ] .

    . g (1 ,2) .

    . g (1) = g (2) = 0 .

    Rolle, (1 ,2) , (1,1) ,

    g () = 0 .

    . , .

    6. Rolle.

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