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Thread: Application of Derivatives

  1. #1
    Senior Member pankaj's Avatar
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    Application of Derivatives

    Let $\displaystyle f:R\to R$ be a twice differentiable function and suppose that for all $\displaystyle x\in R$,$\displaystyle f$ satisfies the following two conditions:

    (i)$\displaystyle |f(x)|\leq 1$

    (ii)$\displaystyle |f''(x)|\leq 1$

    Prove that $\displaystyle |f'(x)|\leq 2$
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  2. #2
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    Quote Originally Posted by pankaj View Post
    Let $\displaystyle f:R\to R$ be a twice differentiable function and suppose that for all $\displaystyle x\in R$,$\displaystyle f$ satisfies the following two conditions:

    (i)$\displaystyle |f(x)|\leq 1$

    (ii)$\displaystyle |f''(x)|\leq 1$

    Prove that $\displaystyle |f'(x)|\leq 2$
    using Taylor's theorem, we know that for any $\displaystyle x \in \mathbb{R}$ there exists some $\displaystyle c \in \mathbb{R}$ such that $\displaystyle f(x+2)=f(x)+2f'(x)+2f''(c).$ therefore:

    $\displaystyle 2|f'(x)|=|f(x+2)-f(x)-2f''(c)| \leq |f(x+2)|+|f(x)|+2|f''(c)| \leq 4,$ and hence $\displaystyle |f'(x)| \leq 2.$


    a similar argument gives us this: if $\displaystyle |f(x)| \leq a$ and $\displaystyle |f''(x)| \leq b,$ for all $\displaystyle x \in \mathbb{R},$ then $\displaystyle |f'(x)| \leq 2 \sqrt{ab},$ for all $\displaystyle x \in \mathbb{R}.$
    Last edited by NonCommAlg; Jun 18th 2009 at 05:55 AM.
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  3. #3
    Senior Member pankaj's Avatar
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    This is being as precise as possible
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