Absolute Maxima and Minima

The absolute maxima and minima describe the largest and smallest values a function takes on an interval.

Definition 7.8. Let $x_0$ be in the domain $D$ of $f(x)$. Then $f(x_0)$ is the absolute maximum value of $f$ on $D$ if $f(x_0)\ge f(x)$ for all $x\in D$; $f(x_0)$ is the absolute minimum value if $f(x_0)\le f(x)$ for all $x\in D$.

There is no general guarantee that an arbitrary function has an absolute maximum or minimum on a given interval (it may run off to $\pm\infty$, or the interval may be open so the "would-be" extreme value is never actually attained). However:

Theorem 7.8 (Extreme Value Theorem). If $f(x)$ is continuous on a closed interval $[a,b]$, then $f$ has both an absolute maximum and an absolute minimum on $[a,b]$.

The absolute extrema occur either at the two endpoints of $[a,b]$ or inside the open interval $(a,b)$; if inside, they must occur at a critical number of $f(x)$. This gives a clean 3-step procedure for finding the absolute extrema of a continuous $f$ on $[a,b]$:

Step 1. Find the critical numbers of $f(x)$ in $(a,b)$.

Step 2. Evaluate $f(x)$ at every critical number and at both endpoints $a,b$. …