1. **State the problem:** We are given the graph of the second derivative $f''$ of a function $f$ on the domain $(-9,9)$ with points of inflection at $x=-3$, $x=0$, and $x=5$. We need to analyze the behavior of $f$, $f'$, and $f''$ on the interval $(7,9)$. 2. **Analyze $f''$ on $(7,9)$:** From the description, $f''$ has a local maximum near $x=7$ and then decreases on $(7,9)$. Since $f''$ crosses the x-axis at $x=5$ and is positive near $7$, on $(7,9)$, $f''$ is positive but decreasing. 3. **Interpret $f''$ behavior:** - $f''$ is **positive** on $(7,9)$, meaning $f$ is **concave up** there. - $f''$ is **decreasing** on $(7,9)$, so the slope of $f'$ is decreasing. 4. **Behavior of $f$ on $(7,9)$:** Since $f''>0$, $f$ is concave up, so the graph of $f$ is curved upward. 5. **Behavior of $f'$ on $(7,9)$:** Since $f''$ is positive but decreasing, $f'$ is increasing but at a decreasing rate. This means $f'$ is increasing but concave down. 6. **Summary:** - The graph of $f''$ on $(7,9)$ is **positive, decreasing, and concave down**. - The graph of $f$ on $(7,9)$ is **concave up** because $f''$ is positive. - The graph of $f'$ on $(7,9)$ is **increasing but concave down** because $f''$ is positive but decreasing. **Final answers:** - $f''$ on $(7,9)$ is positive, decreasing, and concave down. - $f$ on $(7,9)$ is concave up. - $f'$ on $(7,9)$ is increasing but concave down.