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Twice continuously differentiable function

  1. May 24, 2004 #1
    Hello again,

    another problem: given: a function

    [tex] f:[0,\infty)\rightarrow\mathbb{R},f\in C^2(\mathbb{R}^+,\mathbb{R})\\ [/tex]

    The Derivatives

    [tex] f,f''\\ [/tex]

    are bounded.

    It is to proof that

    [tex] \rvert f'(x)\rvert\le\frac{2}{h}\rvert\rvert f\rvert\rvert_{\infty}+\frac{2}{h}\rvert\lvert f''\rvert\rvert_{\infty}\\ [/tex]

    [tex]\forall x\ge 0,h>0\\ [/tex]


    [tex] \rvert\rvert f'\rvert\rvert_{\infty}\le 2(\rvert\rvert f\rvert\rvert_{\infty})^{\frac{1}{2}}(\rvert\rvert f''\rvert\rvert_{\infty})^{\frac{1}{2}}\\ [/tex]

    I began like this:

    [tex] f'(x)=\int_{0}^{x}f''(x)dx\Rightarrow [/tex]

    [tex] \rvert f'(x)\rvert\le\rvert\int_{0}^{x}f''(x)dx\rvert\le\int_{0}^{x}\rvert f''(x)\rvert dx [/tex]

    But then already I don´t know how to go on :yuck:
    I´d be glad to get some hints!

    EDIT: Would it make sense to apply the Tayler series here?
    Last edited: May 24, 2004
  2. jcsd
  3. May 26, 2004 #2
    What is h ?
  4. May 26, 2004 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Don't include x as the variable in your integral and as a limit, it will only confuse you unnecessarily.
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