1. **Problem Statement:** We are given the graph of the derivative function $f'$ of some function $f$. We need to find: - (a) The $x$-values where $f$ has points of inflection. - (b) The intervals where $f$ is concave down. 2. **Recall the concepts:** - Points of inflection of $f$ occur where $f''(x)$ changes sign, equivalently where $f'(x)$ has local maxima or minima (critical points). - $f$ is concave down where $f''(x) < 0$, which means $f'(x)$ is decreasing on those intervals. 3. **Analyze the graph of $f'$:** - $f'$ has a local maximum near $x \approx -0.2$. - $f'$ has a local minimum near $x \approx 3$. 4. **Answers:** - (a) Points of inflection of $f$ occur where $f'$ has local maxima or minima, so $$ x = -0.2, 3 $$ - (b) $f$ is concave down where $f'$ is decreasing. Observing $f'$, it decreases between: $$ (-0.2, 3) $$ 5. **Summary:** - (a) Points of inflection at $x = -0.2$ and $x = 3$. - (b) $f$ is concave down on the interval $( -0.2, 3 )$.