## Calculus (3rd Edition)

$c= ±\sqrt {7}$
The slope of the secant line is $\frac{f(b)-f(a)}{b-a}=\frac{f(5)-f(-4)}{5-(-4)}=\frac{5^{3}-(-4)^{3}}{9}=21$ We must find $c$ such that $f'(c)=21$. $f'(c)=3c^{2}=21\implies c^{2}=\frac{21}{3}=7$ $\implies c= ±\sqrt {7}$ The value $c=±\sqrt {7}$ lies in (-4, 5) and satisfies $f'(±\sqrt {7})=21$. This satisfies the conclusion of the MVT.