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Inequality problem

  1. Dec 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]f\in C^{1}[/tex] on [0,1], and [tex]f(0)=f(1)=0[/tex], prove that
    [tex]\int_{0}^{1}(f(x))^{2}dx \leq \frac{1}{4} \int_{0}^{1} (f'(x))^{2}dx[/tex]

    2. Relevant equations

    3. The attempt at a solution
    what is the trick to produce a 1/4 there? and how to make use of f(0)=f(1)=0? well I know that f(0)=0 gives a formula like [tex]f(x)=\int_{0}^{x} f'(t)dt[/tex] and f(1)=0 gives [tex]f(x)= -\int_{x}^{1} f'(t)dt[/tex]. But seems that I cannot go further.
    Any small hint would be great, thanks!
  2. jcsd
  3. Dec 9, 2009 #2
  4. Dec 9, 2009 #3
    Isn't this Wirtinger's Inequality?
  5. Dec 9, 2009 #4


    Staff: Mentor

  6. Dec 9, 2009 #5
    yea, I'm reading it. cool. Thanks guys!
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