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Let $f:[a,b]\rightarrow R$ be differentiable at $c\in (a,b)$ with $f'(c)<0$. Does this imply that $\exists$$\delta>0$ such that f is monotonic in $(c-\delta,c+\delta)$?

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4 Answers 4

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Define $f$ to be $x+2x^2\sin\frac{1}{x}$ for $x\neq 0$ and $0$ at $x=0$. This is an example of a function that has a positive derivative, but is not monotonic on any open interval surrounding that point. Now just multiply it by $-1$.

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The short answer is: no. Here is an example: $$ f(x) = -x^2(2+\sin(1/x)), \qquad f(0)=0 $$

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Define $f:(-1,1)\rightarrow\mathbb{R}$ by $$ f(x) = \left\{ \begin{array}{rl} -\sin(x) &\mbox{ if $x$ is rational} \\ - x &\mbox{ otherwise} \end{array} \right. $$

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There are even differentiable functions that are not monotone on any interval, as seen in a previous question.

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