Chapter 4 - Applications of the Derivative - 4.3 The Mean Value Theorem and Monotonicity - Exercises - Page 190: 61

$$f(4) \geq 8$$

By using the MVT, there exists $c\in[a,b]$ such that $$ f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} $$ Applying on $[2,4]$ \begin{align*} \frac{f(4)-f(2)}{4-2}&=f^{\prime}(c)\\ \frac{f(4)-(-2)}{2}& \geq 5\\ f(4)&\geq 8 \end{align*}