1. The problem asks us to identify which graph could represent the second derivative $f''(x)$ of a function $f(x)$ given its shape. 2. The original function $f(x)$ starts below the x-axis at $-a$, passes through the origin, rises above the x-axis at $a$, and has a peak and a valley. 3. Recall that $f''(x)$ represents the concavity of $f(x)$: where $f''(x) > 0$, $f(x)$ is concave up (valley), and where $f''(x) < 0$, $f(x)$ is concave down (peak). 4. From the description, $f(x)$ has a peak (concave down) and a valley (concave up). Therefore, $f''(x)$ should be negative at the peak and positive at the valley. 5. The graph that matches this behavior is the one where $f''(x)$ changes sign accordingly: negative near the peak and positive near the valley. 6. Among the options (a), (b), (c), and (d), option (b) shows slopes altering and decreasing downward, consistent with $f''(x)$ changing sign. 7. Hence, the graph in bottom-left-center (b) best represents $f''(x)$ for the given $f(x)$. Final answer: The graph (b) represents $f''(x)$.