1. **Problem Statement:** Given the graph of the second derivative $f''(x)$, determine which statement about the function $f$ is true. 2. **Understanding the problem:** The graph shows $f''(x)$ positive for $x < 0$ and negative for $x > 0$, crossing zero at $x=0$. 3. **Recall key concepts:** - If $f''(x) > 0$, the function $f$ is convex upward (concave up). - If $f''(x) < 0$, the function $f$ is convex downward (concave down). - An inflection point occurs where $f''(x)$ changes sign. - Local maxima or minima depend on the first derivative $f'(x)$, not directly on $f''(x)$. 4. **Analyze each statement:** (a) $f$ is convex upward on $(-,0)$ because $f''(x) > 0$ there. This is true. (b) $f$ has an inflection point at $x=0$ because $f''(x)$ changes from positive to negative at $0$. This is true. (c) $f$ is convex upward on $(0,)$ but $f''(x) < 0$ there, so $f$ is convex downward, so this is false. (d) $f$ has no local maximum value. This cannot be concluded from $f''(x)$ alone. 5. **Conclusion:** Both (a) and (b) are true, but since the question asks which statement is true, the best answer is (b) because it directly relates to the inflection point at $x=0$. **Final answer:** The function $f$ has an inflection point at $x=0$.