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Math Help - Mean value theorem

  1. #1
    Junior Member
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    Mean value theorem

    Hi,

    I must show that \forall n from \mathbb{N}* \forall p from ]0,1[:

    \frac{p}{(n+1)^{1-p}}\le (n+1)^p-n^p \ge \frac{p}{n^{1-p}} by using mean value theorem, can you help me please???
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  2. #2
    MHF Contributor

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    I assume that you mean that \mathbb{N}=\mathbb{Z}^+.
    Define f(x)=x^p then f'(x)=px^{p-1}=\frac{p}{x^{1-p}}.
    Then by mean value \left( {\exists c \in (n,n + 1)} \right)\left[ {f'(c) = f(n + 1) - f(n)} \right] or \frac{p}{c^{1-p}}=(n+1)^p-n^p.
    Note that \frac{1}{n+1}<\frac{1}{c} <\frac{1}{n} .
    Do the algebra to finish.
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