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Thread: Differentiable function

  1. #1
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    Differentiable function

    Suppose $\displaystyle f(x)$ is a diferentiable function and $\displaystyle g(x)$ is a doubly differentiable function such that $\displaystyle f'(x)=g(x)$ and $\displaystyle |f(x)| \leq 1$ and $\displaystyle |g(x)| \geq 2$ for all $\displaystyle x \in [-3,3]$. If further $\displaystyle [f(0)]^{2}+[g(0)]^{2}=9$, prove that there exists some $\displaystyle c \in (-3,3)$ such that $\displaystyle g(c) \cdot g''(c)<0$.
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  2. #2
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    Sorry a Mistake

    There is no condition on $\displaystyle g$ as $\displaystyle |g(x)|>2$.
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