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My text book contains the following task which I'm unsure of:

Be $f: [a, b] \rightarrow \mathbb{R}$ differentiable in $b$ and $f\;'(b)>0$. Prove that $f$ contains an isolated local maximum at $b$ (this means there is a $\delta > 0$ with $f(b) > f(x)$ for all $x \in (b- \delta, b)$).

However to my understanding the derivative in $b$ has to be 0 as it contains a maximum at $b$ and the slope is zero. Can it be that this is a error in the book and $f''(b)<0$ or $f(b)>0$ is meant or am I missing something?

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    If thinks overlap you can add \; to put a bit of space in. I did it for your f'.2011-06-29
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    Do you remember the exact hypotheses that you need in order to conclude that a local maximum $x_0$ has $f\,'(x_0) = 0$? Also, consider $f(x) = x^2$ on $[a,b] = [0,1]$ as an example supporting the task.2011-06-29
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    I am not a native speaker of English, but I think derivative instead of derivation should be in the title. Or am I wrong?2011-06-29
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    @Martin: Only if he is asking not about derivation the proof about derivative.2011-06-29

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