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?