1. The problem asks us to identify which graph could represent the second derivative $f''(x)$ of the function $f(x)$ shown. 2. The graph of $f(x)$ crosses the x-axis at $-a$, $0$, and $a$, with a peak between $-a$ and $0$ and a trough between $0$ and $a$. 3. Recall that the second derivative $f''(x)$ indicates the concavity of $f(x)$: - Where $f(x)$ is concave up, $f''(x) > 0$. - Where $f(x)$ is concave down, $f''(x) < 0$. 4. From the graph of $f(x)$: - Between $-a$ and $0$, the curve is concave down (peak), so $f''(x) < 0$ there. - Between $0$ and $a$, the curve is concave up (trough), so $f''(x) > 0$ there. 5. The zeros of $f''(x)$ correspond to inflection points of $f(x)$, where concavity changes. These occur near $x=0$. 6. Therefore, $f''(x)$ should be negative on $(-a,0)$, zero at $0$, and positive on $(0,a)$, crossing the x-axis at $0$. 7. Among the given options, graph (b) shows a function that is negative on the left side of zero and positive on the right side, crossing zero at the origin. 8. Hence, graph (b) best represents $f''(x)$. Final answer: The graph labeled (b) represents $f''(x)$.