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Math Help - Application of derivatives

  1. #1
    Senior Member pankaj's Avatar
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    Application of derivatives

    Let f(x) be a thrice continuously differentiable function.Prove or disprove that between two consecutive roots of f'''(x)=0 there exist atmost 4 roots of f(x)=0
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    Quote Originally Posted by pankaj View Post
    Let f(x) be a thrice continuously differentiable function.Prove or disprove that between two consecutive roots of f'''(x)=0 there exist atmost \color{red}4 roots of f(x)=0
    it's just a simple result of Rolle's theorem ... and i think that 4 should be actually 3.
    Last edited by NonCommAlg; November 15th 2008 at 09:04 AM.
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  3. #3
    Senior Member pankaj's Avatar
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    I had drawn the graph and I found that there are indeed 3 points but I am not able to apply Rolle's theorem.I doubt if converse of Rolle's theorem is true.
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    suppose \alpha < \beta are two consecutive zeros of f'''. suppose f has 4 zeros a_i, \ 1 \leq i \leq 4, such that \alpha \leq a_1 < a_2 < a_3 < a_4 \leq \beta. then by Rolle's theorem there exist a_i < b_i < a_{i+1}, \ 1 \leq i \leq 3,

    such that f'(b_i)=0, \ i=1,2,3. again by Rolle's theorem there exist b_i < c_i < b_{i+1}, \ i=1,2, such that f''(c_i)=0, \ i=1,2. finally, again, by Rolle's theorem there exists c_1 < \gamma < c_2 such

    that f'''(\gamma)=0. but obviously \alpha < \gamma < \beta, which contradicts our assumption that \alpha, \beta are two consecutive zeros of f'''. so f can have at most 3 zeros between \alpha, \beta. \ \ \Box
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  5. #5
    Senior Member pankaj's Avatar
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    Somehow I was missing point you made out inthe last sentence.
    Nice proof.Graphical method on which I was working was not very convincing.
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