Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 189: 5

$ 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.