1. **Problem Statement:** We are given the graph of the derivative $f'$ of a twice-differentiable function $f$ on the domain $(-9,9)$. We want to analyze the behavior of $f$, $f'$, and $f''$ on the interval $(6,9)$ based on the graph of $f'$. 2. **Given Information:** - $f'$ is positive and decreasing on $(6,9)$. - Points of inflection of $f'$ occur at $x=-4,0,2,5$. 3. **Recall Important Concepts:** - $f'(x)$ is the slope of $f(x)$. - $f''(x)$ is the derivative of $f'(x)$, indicating the concavity of $f$. - If $f'(x) > 0$, then $f$ is increasing. - If $f'(x)$ is decreasing, then $f''(x) < 0$. 4. **Analyze $f'$ on $(6,9)$:** - Since $f'(x) > 0$, $f$ is increasing on $(6,9)$. - Since $f'(x)$ is decreasing, $f''(x) < 0$ on $(6,9)$. 5. **Summarize the behavior:** - $f'$ is positive and decreasing. - $f$ is increasing because $f' > 0$. - $f''$ is negative because $f'$ is decreasing. **Final answers:** - The graph of $f'$ on $(6,9)$ is **positive**, **decreasing**, and **concave down** (since $f''<0$). - On $(6,9)$, $f$ is **increasing** because $f' > 0$. - On $(6,9)$, $f''$ is **negative** because $f'$ is decreasing.