Answer
(a) $f$ is decreasing on $(1,5)$ and $(7,8)$ and increasing on $(0,1)$ and $(5,7)$.
(b) $f$ has a local minimum at $x=5$ and 2 local maximums at $x=1$ and $x=7$.
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 $(1,5)$ and $(7,8)$, $f'\lt0$. Therefore, $f$ is decreasing on these two intervals.
On $(0,1)$ and $(5,7)$, $f'\gt0$. So $f$ is increasing on these two intervals.
(b) We use The First Derivative Test:
Here, $f'$ changes from negative to positive at $x=5$, so $f$ has a local minimum at $x=5$.
Also, $f'$ changes from positive to negative at $x=1$ and $x=7$, so $f$ has 2 local maximums at $x=1$ and $x=7$.
(Do not let the fluctuations in this graph confuse you. If you look at the definitions, you see that only the sign of $f'$ and whether it changes sign or not matters.)
