1. The problem involves identifying the derivative function $f'(x)$ of a cubic function $f(x)$ shown in the top-right graph. 2. The top-right graph shows $f(x)$, a cubic curve increasing to a maximum near $x=-1$ and then decreasing after $x=0$. The derivative $f'(x)$ should be zero at local maxima and minima of $f(x)$. 3. The bottom graphs (a), (b), (c), and (d) are candidates for $f'(x)$. 4. Since $f(x)$ has a maximum near $x=-1$, $f'(x)$ must be zero near $x=-1$. Also, $f(x)$ decreases after $x=0$, so $f'(x)$ should be negative after $x=0$. 5. Graph (a) is a line with negative slope passing through $(0,2)$ and $(2,0)$, so $f'(0)=2$, which is positive, contradicting the decrease of $f(x)$ after $x=0$. 6. Graph (b) is a line with positive slope passing through $(-2,-2)$ and $(2,2)$, so $f'(x)$ is positive after $x=0$, contradicting the decrease of $f(x)$. 7. Graph (c) is a downward-opening parabola with vertex near $(0,3)$, so $f'(0)=3$ (positive), contradicting the decrease of $f(x)$ after $x=0$. 8. Graph (d) is a line with negative slope passing through $(0,1)$ and $(2,-2)$, so $f'(0)=1$ (positive), but it decreases after $x=0$ and crosses zero near $x=-1$, matching the behavior of $f'(x)$. 9. Therefore, the derivative $f'(x)$ corresponds to graph (d). 10. The equation of graph (d) can be found using points $(0,1)$ and $(2,-2)$: $$m=\frac{-2-1}{2-0} = \frac{-3}{2} = -1.5$$ $$f'(x) = -1.5x + 1$$ Final answer: The derivative function is $$f'(x) = -\frac{3}{2}x + 1$$.