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Math Help - An Integral Inequality #1

  1. #1
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    Arrow An Integral Inequality #1

    Assume f \in C^{(1)} ([0,1]),and f(0)=0,f(1)=1, then


    \int_0^1 |f(x)-f'(x)| dx \geq \frac{1}{e}


    thanks very much
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  2. #2
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    Quote Originally Posted by Xingyuan View Post
    Assume f \in C^{(1)} ([0,1]),and f(0)=0,f(1)=1, then


    \int_0^1 |f(x)-f'(x)| dx \geq \frac{1}{e}


    thanks very much
    Use the fact that \forall x \in [0,1], |e^{-x}(f(x) - f'(x))| \leq |f(x) - f'(x)|

    \Rightarrow \int^1_0 |e^{-x}(f(x) - f'(x))|\, dx \leq \int^1_0|f(x) - f'(x)|\, dx

    \Rightarrow \int^1_0 d(|e^{-x}f(x)|) \leq \int^1_0|f(x) - f'(x)|\, dx

    \Rightarrow |e^{-x}f(x)|\bigg{|}_{0}^{1} \leq \int^1_0|f(x) - f'(x)|\, dx

    \Rightarrow |e^{-1}f(1)| - |e^{-0}f(0)| \leq \int^1_0|f(x) - f'(x)|\, dx

    \Rightarrow  \int^1_0|f(x) - f'(x)|\, dx \geq \frac1{e}
    Last edited by Isomorphism; May 5th 2008 at 01:43 AM.
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