Question: The derivative of the twice-differentiable function $f$ is shown below on the domain $(-9, 9)$. The graph of $f'$ has points of inflection at $x = -5$, $x = -1$, $x = 3$, indicated by small green circles. What inferences can be made about the graphs of $f$, $f'$, and $f''$ when $x = -6$?

1. **State the problem:** 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 $x=-5$, $x=-1$, and $x=3$. We want to infer the behavior of $f$, $f'$, and $f''$ at $x=-6$. 2. **Recall definitions and relationships:** - $f'(x)$ is the derivative of $f(x)$, representing the slope of $f$ at $x$. - $f''(x)$ is the derivative of $f'(x)$, representing the concavity of $f$ or the slope of $f'$ at $x$. - Points of inflection of $f'$ correspond to zeros of $f''$ or changes in concavity of $f'$. 3. **Analyze the graph of $f'$ at $x=-6$:** - The graph of $f'$ at $x=-6$ is in the top-left region, above the $x$-axis, near a local maximum with $f'(-6) \approx 7$. - Since $f'(-6) > 0$, the function $f$ is increasing at $x=-6$. - Because $f'$ is near a local maximum at $x=-6$, the slope of $f'$ (which is $f''$) changes from positive to negative, so $f''(-6) < 0$. 4. **Summarize the inferences:** - $f$ is increasing at $x=-6$ because $f'(-6) > 0$. - $f'$ has a local maximum at $x=-6$, so $f'$ is positive and decreasing there. - $f''$ is negative at $x=-6$ because $f'$ is decreasing at that point. 5. **Final answer:** - Based on the graph, when $x = -6$, the graph of $f'$ is positive and near a local maximum. - The function $f$ is increasing and has positive slope. - The second derivative $f''$ is negative at $x = -6$. Thus: - $f$ has **increasing** and has **positive slope**. - $f''$ is **negative** when $x = -6$.