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Twice differentiable functions

  1. Aug 12, 2008 #1
    Hi all. Having a little trouble on this week's problem set. Perhaps one of you might be able to provide some insight.

    1. The problem statement, all variables and given/known data

    [tex] f:[a,b] \rightarrow \mathbb{R} [/tex] is continuous and twice differentiable on (a,b). If f(a)=f(b)=0 and f(c) > 0 for some [tex] c \in (a,b) [/tex] then [tex] \exists \gamma \in (a,b) [/tex] s.t. [tex] f \prime \prime (\gamma) < 0 [/tex]

    2. Relevant equations
    Rolle's Theorem, MVT, Intermediate Value Theorem

    3. The attempt at a solution

    I'm not really sure how to approach the problem. I'm assuming you would apply Rolle's theorem or the Mean Value Theorem, or perhaps the Intermediate Value Property to the problem in order to obtain the solution.

    With those theorems we can infer that [tex]\exists \theta [/tex] st [tex] f \prime (\theta) = 0 [/tex]
    f(c) > 0 so there must be points u,v st f'(u) > 0 and f'(v) < 0

    but none of that really gives me any info on the second derivative, which is what I need. I'm sure I'm overlooking something simple as usual, but if anyone could point me in the right direction it would be greatly appreciated.
  2. jcsd
  3. Aug 12, 2008 #2


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    Science Advisor
    Homework Helper

    Look at the MVT on the intervals [a,c] and [c,b]. So, yes, there is a u in (a,c) and a v in (c,b) such that f'(u)>0 and f'(v)<0. Now apply the MVT to f'(x) on the interval [u,v]. The second derivative is the derivative of the first.
  4. Aug 12, 2008 #3
    thank you very much Dick. You've been extremely helpful as always. I was able to get the answer.
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