Single Variable Calculus, Chapter 4, 4.1, Section 4.1, Problem 2

If $f$ is a continuous function defined on a closed interval $[a, b ]$.

a.) Determine the theorem that guarantees the existence of an absolute maximum value and an absolute minimum value for $f$.

Extreme value theorem. If $f$ is continuous on a closed interval $[a, b]$ then $f$ becomes an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in $[a, b]$

b.) Give the stops to find those maximum and minimum values.

First, determine all values of $x$ for which $f'(x) = 0$ or $f'(x)$ is undefined. Then evaluate $f(x)$ to those values of $x$ as well as the end points. And lastly compare the values of $f(x)$ to find the largest and smallest value.