1. The problem asks to find intervals where the function $f(x)$ is concave up on the open interval $(-9,9)$. 2. Recall that concavity of $f$ is determined by the sign of the second derivative $f''(x)$: - If $f''(x) > 0$, then $f$ is concave up on that interval. - If $f''(x) < 0$, then $f$ is concave down on that interval. 3. Since $f'(x)$ is the first derivative of $f$, the concavity depends on whether $f'(x)$ is increasing or decreasing: - $f''(x) = (f'(x))'$ - If $f'(x)$ is increasing, then $f''(x) > 0$ and $f$ is concave up. - If $f'(x)$ is decreasing, then $f''(x) < 0$ and $f$ is concave down. 4. From the description of the graph of $f'(x)$ (orange curve): - $f'(x)$ decreases from $x = -9$ to about $x = -1$. - $f'(x)$ increases from about $x = -1$ to $x = 9$. 5. Therefore, $f''(x) > 0$ (concave up) where $f'(x)$ is increasing, which is approximately on the interval $$(-1,9)$$. 6. Final answer: The function $f(x)$ is concave up on the open interval $$\boxed{(-1,9)}$$.