1. The problem states that the given graph is the first derivative $f'(x)$ of a function $f(x)$, and we need to determine which of the provided graphs could represent the original function $f(x)$. 2. Important rules: - Where $f'(x) > 0$, the function $f(x)$ is increasing. - Where $f'(x) < 0$, the function $f(x)$ is decreasing. - Where $f'(x) = 0$, $f(x)$ has critical points (possible maxima, minima, or inflection points). 3. The given $f'(x)$ is a U-shaped curve opening upwards with zeros at $x = -1$ and $x = 1$, and $f'(x) o ext{positive}$ outside these points. 4. Analyze $f'(x)$ sign: - For $x < -1$, $f'(x) > 0$ so $f(x)$ is increasing. - For $-1 < x < 1$, $f'(x) < 0$ so $f(x)$ is decreasing. - For $x > 1$, $f'(x) > 0$ so $f(x)$ is increasing. 5. This means $f(x)$ has a local maximum at $x = -1$ and a local minimum at $x = 1$. 6. Among the options: - Graph (a) crosses x-axis three times and oscillates, not matching the shape. - Graph (b) crosses x-axis twice, but shape is a single wave, not matching the increasing-decreasing-increasing pattern. - Graph (c) is W-shaped crossing x-axis three times, matching the pattern of increasing, decreasing, increasing. - Graph (d) is a straight line, which cannot have a U-shaped derivative. 7. Therefore, the graph (c) is the general shape of $f(x)$ corresponding to the given $f'(x)$. Final answer: Graph (c)