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Math Help - Continuous function Mean value theorem

  1. #1
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    Continuous function Mean value theorem

    Suppose that
    f : [0, 1]--->R is continuously differentiable.

    (a) Show that there is some number
    M such that |f'(x)|< M for all x.

    I understand mean value therorem but use it to solve this problem. Can you give me some hints please?

    Thanks
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  2. #2
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    A continuously differentiable function on a compact set means that it's continuous (in this case) on [0,1] and with a continuous derivative on [0,1]. Hence, since the derivative is continuous on a compact set, it's bounded.
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  3. #3
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    It seems to me like this would require the extreme value theorem (applied to f'), not the mean value theorem, but maybe I am just missing something.
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  4. #4
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    From lecture notes: By the boundedness principle, a continuous function on a closed interval attains its maximum and minimum.

    Therefore there exists a number M such that f(x)<=M for all x belong to [0,1]. How do i apply this to |f'|

    I think i have to do something like.... how do i complete the answer? thanks
    0<|f'|<max(m1,m2)
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  5. #5
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    Quote Originally Posted by charikaar View Post
    From lecture notes: By the boundedness principle, a continuous function on a closed interval attains its maximum and minimum.

    Therefore there exists a number M such that f(x)<=M for all x belong to [0,1]. How do i apply this to |f'|

    I think i have to do something like.... how do i complete the answer? thanks
    0<|f'|<max(m1,m2)
    You said "f : [0, 1]--->R is continuously differentiable." "Continuously differentiable" (as opposed to just "differentiable") means that f' is also continuous. Apply the boundedness principle to f', not f.
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