1. **Problem Statement:** Given the graph of the derivative function $f'(x)$ of a continuous function $f$ on $\mathbb{R}$, determine which statement about $f$ is true. 2. **Understanding the graph:** The graph of $f'(x)$ is a parabola opening upwards with vertex at $(0,0)$. - It crosses the x-axis at $x=-1$ and $x=1$. - $f'(x) < 0$ on the interval $(-1,1)$. - $f'(x) > 0$ outside the interval $(-1,1)$. 3. **Recall key facts:** - $f'(x) = 0$ means $f$ has a horizontal tangent at $x$. - If $f'(x) > 0$ on an interval, $f$ is increasing there. - If $f'(x) < 0$ on an interval, $f$ is decreasing there. - If $f'(x)$ changes from positive to negative at $x=c$, $f$ has a local maximum at $c$. - If $f'(x)$ changes from negative to positive at $x=c$, $f$ has a local minimum at $c$. 4. **Analyze each statement:** (a) At $x=0$, $f'(0) = 0$ (vertex), so $f$ has a horizontal tangent at $x=0$. This is true. (b) On $(-1,1)$, $f'(x) < 0$, so $f$ is decreasing there. This is true. (c) At $x=1$, $f'(x)$ changes from negative (just left of 1) to positive (just right of 1), so $f$ has a local minimum at $x=1$, not a maximum. This is false. (d) At $x=-1$, $f'(x)$ changes from positive (left of -1) to negative (right of -1), so $f$ has a local maximum at $x=-1$, not a minimum. This is false. 5. **Conclusion:** Statements (a) and (b) are true, but since the question asks which is true (singular), the best answer is (a) because it is a direct consequence of $f'(0)=0$. **Final answer:** The function $f$ has a horizontal tangent at $x=0$.