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Math Help - find the max value of integral

  1. #1
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    Question find the max value of integral

    If f: \mathbb{R}\mapsto\mathbb{R} is continuous, and \int^1_0\left(f(x)\right)^2 dx = 3.
    Find the maximum value of \int^1_0 x f(x) dx

    Please explain to me...
    You are so nice^^
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  2. #2
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    Quote Originally Posted by GTK X Hunter View Post
    If f: \mathbb{R}\mapsto\mathbb{R} is continuous, and \int^1_0\left(f(x)\right)^2 dx = 3.
    Find the maximum value of \int^1_0 x f(x) dx

    Please explain to me...
    You are so nice^^
    by Holder's inequality: \int_0^1 xf(x) \ dx \leq \int_0^1 x|f(x| \ dx \leq \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1. also if f(x) = 3x, then \int_0^1 (f(x))^2 dx = 3 and \int_0^1 xf(x) dx = 1. thus \max_f \int_0^1 xf(x) \ dx = 1.
    Last edited by NonCommAlg; October 13th 2009 at 09:11 PM.
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  3. #3
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    Quote Originally Posted by GTK X Hunter View Post
    If f: \mathbb{R}\mapsto\mathbb{R} is continuous, and \int^1_0\left(f(x)\right)^2 dx = 3.
    Find the maximum value of \int^1_0 x f(x) dx

    Please explain to me... explain to me..
    Hint: Did you ever come across the Cauchy–Schwarz inequality?
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  4. #4
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    Quote Originally Posted by NonCommAlg View Post
    by Holder's inequality: \int_0^1 xf(x) \ dx \leq \int_0^1 x|f(x| \ dx \leq \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1. also if f(x) = 3x, then \int_0^1 (f(x))^2 dx = 3 and \int_0^1 xf(x) dx = 1. thus \max_f \int_0^1 xf(x) \ dx = 1.
    How can you conclude that \sqrt{\int_0^1 x^2 dx \cdot \int_0^1 (f(x))^2 dx} = 1.
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  5. #5
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    first assumption was \int^1_0\left(f(x)\right)^2 dx = 3, and \int_0^1x^2\,dx=\frac13.
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