(a) $f$ is decreasing on $(0,1)$ and $(5,6)$ and increasing on $(1,5)$. (b) $f$ has a local minimum at $x=1$ and a local maximum at $x=5$.
Work Step by Step
(a) We use the Increasing/Decreasing Test: Therefore, looking at the graph, we would look for the intervals where $f'$ is negative and positive. We see that on $(0,1)$ and $(5,6)$, $f'\lt0$. Therefore, $f$ is decreasing on these two intervals. On $(1,5)$, $f'\gt0$. So $f$ is increasing on this interval. (b) We use The First Derivative Test: Here, $f'$ changes from negative to positive at $x=1$, so $f$ has a local minimum at $x=1$. Also, $f'$ changes from positive to negative at $x=5$, so $f$ has a local maximum at $x=5$.