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Homework Help: Prove that f(100) < 100

  1. Aug 26, 2011 #1
    1. The problem statement, all variables and given/known data


    For f(0) = 0, and that f'(x) = [tex]\frac{1}{1 + e^{-f(x)}}[/tex], prove that f(100) < 100


    3. The attempt at a solution

    I did

    [tex]\f(100) = \int_{0}^{100} \frac{dx}{1 + e^{-f(x)}}[/tex]

    Unfortunately, I got f(100) back...
     
  2. jcsd
  3. Aug 26, 2011 #2

    fzero

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  4. Aug 26, 2011 #3
    That's already ok. Can you prove now that the integral must be <100??

    First, can you prove that

    [tex]\frac{1}{1+e^{-f(x)}}\leq 1[/tex]
     
  5. Aug 26, 2011 #4

    Ray Vickson

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    For y > = 0 we have 1/(1+exp(-y)) = exp(y)/[1+exp(y)] <= 1.

    RGV
     
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