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Math Help - Cauchy-Schwarz inequality

  1. #1
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    Cauchy-Schwarz inequality

    Hi, I am stuck on this question.

    Prove

    |f(L)-f(0)|^2 \le L \int_0^L |f'(x)|^2 dx for any function f \in C^1([0,L]).

    So I know I should use the Cauchy-Schwarz inequality applied to the functions f′ and 1, but I am stuck on how to do it.
    Thanks for any help people.
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  2. #2
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    Re: Cauchy-Schwarz inequality

    |f(L)-f(0)|=|\int_0^L f'(x)dx|\leq \int_0^L |f(x)|\cdot 1 dx\leq (\int_0^L|f'(x)|^2dx)^{1/2}\cdot (\int_0^L 1^2dx)^{1/2}
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  3. #3
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    Re: Cauchy-Schwarz inequality

    Thanks for that, but how do I go on to prove my inequality? I don't see how squaring brings the L in.
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  4. #4
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    Re: Cauchy-Schwarz inequality

    Quote Originally Posted by iceman954 View Post
    Thanks for that, but how do I go on to prove my inequality? I don't see how squaring brings the L in.
    What is

    \int_{0}^{L}1^{2}\,dx?
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