1. **Problem statement:** We are given the graph of the derivative $f'$ and asked to find: (a) The $x$-values where the points of inflection of $f$ occur. (b) The intervals where $f$ is concave down. 2. **Recall definitions:** - Points of inflection of $f$ occur where the concavity changes, i.e., where the second derivative $f''$ changes sign. - Since $f' = \frac{df}{dx}$, the second derivative $f'' = \frac{d}{dx}f'$ is the slope of the graph of $f'$. - Points of inflection correspond to where the slope of $f'$ changes from positive to negative or negative to positive. 3. **Analyze the graph of $f'$:** - The graph of $f'$ starts near zero at $x=0$, increases to a peak near $x=5$, then decreases steeply passing below the $x$-axis before rising again near $x=10$. - The slope of $f'$ changes at points where $f'$ has local maxima or minima. - From the description, the slope changes at approximately $x=0$ and $x=10$ where $f'$ crosses the $x$-axis and the slope changes. 4. **Answer (a):** - Points of inflection of $f$ occur at $x=0$ and $x=10$. 5. **Concavity of $f$:** - $f$ is concave down where $f'' < 0$, i.e., where the slope of $f'$ is negative. - From the graph, $f'$ slopes downward between $x=0$ and $x=9$ (approximately), so $f$ is concave down on $(0,9)$. 6. **Answer (b):** - $f$ is concave down for $x$ in the interval $(0,9)$. **Final answers:** (a) $x=0,10$ (b) $f$ is concave down on $(0,9)$