MATH 144: Calculus for the Physical Sciences I

Concept 6.3.1. Min and max of a function.

Let $c \in D$ where $D$ is the domain of $f\text{.}$ Then $f(c)$ is:

• The absolute maximum of $f$ on $D$ if $f(c) \geq f(x)$ for all $x \in D\text{;}$
• The absolute minimum of $f$ on $D$ if $f(c) \leq f(x)$ for all $x \in D\text{;}$
• The local maximum of $f$ on $D$ if $f(c) \geq f(x)$ for all $x$ near $c\text{;}$
• The local minimum of $f$ on $D$ if $f(c) \leq f(x)$ for all $x$ near $c\text{.}$

Near $c$ means “for all $x$ in some open interval containing $c\text{.}$”

In general, absolute minima and maxima can occur either at local minima or maxima, or, if $D$ is a closed interval, at the endpoints of the interval.

Concept 6.3.2. Extrema and critical numbers.

A critical number of $f$ is a number $c$ in the domain $D$ of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist.

If $f$ has a local minimum or maximum at $c\text{,}$ then $c$ is a critical number of $f$ (Fermat's Theorem). However, the converse is not true: not all critical numbers are local minima or maxima.

Concept 6.3.3. Extreme Value Theorem.

If $f$ is continuous on $[a,b]\text{,}$ then $f$ must attain an absolute maximum $f(c)$ and an absolute minimum $f(d)$ at some numbers $c,d \in [a,b]\text{.}$

Concept 6.3.4. How to find the absolute extrema of a continuous function $f$ on a closed interval $[a,b]$.

1. Find the critical numbers of $f$ in $(a,b)\text{,}$ and evaluate $f$ at the critical numbers;
2. Evaluate $f$ at the endpoints $a$ and $b\text{;}$
3. Compare the values. The largest of these values is the absolute maximum, while the smallest is the absolute minimum.