1. **Problem Statement:** We are given the graph of the first derivative $f'(x)$ of a function $f(x)$, which is a parabola opening upwards with roots at $x=-1$ and $x=1$. We need to determine which of the given graphs (a, b, c, or d) could represent the original function $f(x)$. 2. **Understanding the derivative:** The first derivative $f'(x)$ is zero at $x=-1$ and $x=1$, and positive or negative elsewhere depending on the parabola shape. 3. **Key formula and rules:** - The roots of $f'(x)$ correspond to critical points (local maxima, minima, or inflection points) of $f(x)$. - Since $f'(x)$ is a parabola opening upwards with roots at $-1$ and $1$, it can be expressed as: $$f'(x) = a(x+1)(x-1) = a(x^2 - 1)$$ where $a > 0$ because it opens upwards. 4. **Sign analysis of $f'(x)$:** - For $x < -1$, $f'(x) > 0$ (since both $(x+1)$ and $(x-1)$ are negative, their product is positive). - For $-1 < x < 1$, $f'(x) < 0$ (one factor positive, one negative). - For $x > 1$, $f'(x) > 0$. 5. **Implications for $f(x)$:** - $f(x)$ is increasing when $f'(x) > 0$ and decreasing when $f'(x) < 0$. - Therefore, $f(x)$ increases on $(-, -1)$, decreases on $(-1, 1)$, and increases again on $(1, )$. - This means $f(x)$ has a local maximum at $x=-1$ and a local minimum at $x=1$. 6. **Checking the options:** - (a) and (b) describe cubic-like functions with two turning points and three x-intercepts, consistent with local max and min. - (c) describes a cubic-like function with a single minimum between 0 and 1, which does not match the critical points at $-1$ and $1$. - (d) is a straight line, which cannot have a derivative that is a parabola. 7. **Conclusion:** The function $f(x)$ must have two turning points at $x=-1$ and $x=1$, with a local maximum at $-1$ and a local minimum at $1$. Both (a) and (b) fit this description, but since the problem states the derivative is positive outside $[-1,1]$ and negative inside, the shape with a local max at $-1$ and local min at $1$ matches these options. **Final answer:** The graphs labeled (a) and (b) could represent the general shape of $f(x)$ given the derivative $f'(x)$.