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Math Help - Application of Derivatives

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

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

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

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

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

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

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

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

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


    a similar argument gives us this: if |f(x)| \leq a and |f''(x)| \leq b, for all x \in \mathbb{R}, then |f'(x)| \leq 2 \sqrt{ab}, for all x \in \mathbb{R}.
    Last edited by NonCommAlg; June 18th 2009 at 06:55 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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