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Math Help - Continuity/Differentiability Question

  1. #1
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    Continuity/Differentiability Question

    Suppose f:[a,b] \rightarrow \mathbb{R} is differentiable on [a,b] (with one sided derivatives at the end points). Show that if f\prime(a)<0<f\prime(b) then the minimum of f is attained at a point c \in (a,b). Note: You may not assume the derivative of f is continuous.

    My first thoughts was just to use IVT on the derivative until I saw the note, now I'm not sure what to do. Any help where to start?
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  2. #2
    Super Member girdav's Avatar
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    We know that the minimum is attained at a point c\in \left[a,b\right]. We only have to show that it can't be at a or b. If the minimum is attained at a for example then exists h_0>0 such that if 0<h<h_0 then \frac{f(a+h)-f(a)}h<0. Now, don't forgive that h is positive in order to find a contradiction.
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  3. #3
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    Quote Originally Posted by girdav View Post
    \frac{f(a+h)-f(a)}h<0
    Is this inequality the wrong way round? Should it not be >?
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  4. #4
    Super Member girdav's Avatar
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    Quote Originally Posted by worc3247 View Post
    Is this inequality the wrong way round? Should it not be >?
    No, since f'(a)<0. Then I use the definition of a limit.
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    Ok thanks
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  6. #6
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    Re: Continuity/Differentiability Question

    And just so you know, the IVT property also holds for derivatives, regardless of differentiability.
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