Answer
(a). $-4$,
(b). $-13$,
(c). $-1$.
Work Step by Step
(a). From $x_1=0$ to $x_2=2$, we have the average rate of change as $\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{(-(2)^3+1)-(-(0)^3+1)}{2-0}=\frac{-8}{2}=-4$,
(b). From $x_1=1$ to $x_2=3$, we have the average rate of change as $\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{(-(3)^3+1)-(-(1)^3+1)}{3-1}=\frac{-26}{2}=-13$,
(c). From $x_1=-1$ to $x_2=1$, we have the average rate of change as $\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{(-(1)^3+1)-(-(-1)^3+1)}{1+1}=\frac{-2}{2}=-1$.
