1. **Problem Statement:** We are given the graph of the derivative function $f'(x)$ of a continuous function $f$ on $\mathbb{R}$. The graph is a parabola opening upward with vertex at the origin $(0,0)$ and roots at $x=-1$ and $x=1$. We need to determine which of the statements (a), (b), (c), or (d) about $f$ is true. 2. **Recall the meaning of $f'(x)$:** - $f'(x)$ represents the slope of the tangent line to the graph of $f$ at $x$. - If $f'(x) = 0$, then $f$ has a horizontal tangent at $x$. - If $f'(x) > 0$, then $f$ is increasing at $x$. - If $f'(x) < 0$, then $f$ is decreasing at $x$. 3. **Analyze the graph of $f'(x)$:** - The parabola $f'(x)$ opens upward with vertex at $(0,0)$. - $f'(0) = 0$, so $f$ has a horizontal tangent at $x=0$. - The roots are at $x=-1$ and $x=1$, so $f'(-1) = 0$ and $f'(1) = 0$. - Since the parabola opens upward and crosses the x-axis at $-1$ and $1$, $f'(x)$ is negative between $-1$ and $1$ and positive outside this interval. 4. **Interpret the statements:** - (a) $f$ has a horizontal tangent at $x=0$: True, because $f'(0) = 0$. - (b) $f$ is decreasing on $(-1,1)$: True, because $f'(x) < 0$ for $x$ in $(-1,1)$. - (c) $f$ has a local maximum at $x=1$: Since $f'(1) = 0$ and $f'(x)$ changes from negative to positive at $x=1$, $f$ has a local minimum at $x=1$, not a maximum. - (d) $f$ has a local minimum at $x=-1$: Since $f'(x)$ changes from positive to negative at $x=-1$, $f$ has a local maximum at $x=-1$, not a minimum. 5. **Conclusion:** - Statements (a) and (b) are true. - Statement (c) is false. - Statement (d) is false. **Final answer:** The true statements are (a) and (b).