1. **Problem Statement:** We are given the graph of the derivative $f'$ of a twice-differentiable function $f$ on the domain $(-9,9)$, with points of inflection at $x=-5$, $x=-2$, and $x=1$. We need to analyze the behavior of $f$, $f'$, and $f''$ on the interval $(3,8)$. 2. **Recall definitions and relationships:** - $f'(x)$ is the derivative of $f(x)$, so it represents the slope of $f$. - $f''(x)$ is the derivative of $f'(x)$, so it represents the concavity of $f$ or the slope of $f'$. - Points of inflection on $f'$ correspond to where $f''$ changes sign. 3. **Analyze $f'$ on $(3,8)$:** From the graph description, on $(3,8)$, $f'$ is: - **Positive** (above the $x$-axis), so $f'(x) > 0$. - **Decreasing** (the graph of $f'$ slopes downward). - **Concave up** or **concave down**? Since points of inflection are at $x=1$ and none between $3$ and $8$, the concavity remains consistent. Given the graph is decreasing and no inflection points, $f'$ is likely **concave down** (since slope is decreasing and curvature is downward). 4. **Conclusions about $f$ on $(3,8)$:** - Since $f'(x) > 0$, $f$ is **increasing** on $(3,8)$. 5. **Conclusions about $f''$ on $(3,8)$:** - Since $f'$ is decreasing, $f''(x) = (f')'(x) < 0$ on $(3,8)$. - So $f''$ is **negative** on $(3,8)$. **Final answers:** - The graph of $f'$ on $(3,8)$ is **positive**, **decreasing**, and **concave down**. - On $(3,8)$, $f$ is **increasing** because $f' > 0$. - On $(3,8)$, $f''$ is **negative** because $f'$ is decreasing.