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    Real Analysis : Derivatives

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    If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that:

    f(x) <= 1/2 if x <= 1

    © BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf

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    |f''(x)| <= 1 for |x| <= 1
    => -1 <= f''(x) <= 1 for -1<=x<=1
    integrate from 0 to x:
    => -x <= f'(x) <= x for -1<=x<=1 (because ...

    Solution Summary

    Integration is used to prove a function is true. The absolute value of x being less or equal to one is computed.