Section 3.1 Maximum and Minimum Values

Math 1231: Single-Variable Calculus

Absolute Max and Min

Absolute Max and Min

Absolute max and min are also called global max and min. Both are called extreme values of f(x).

Local Max and Min

Examples

Examples

Extreme Value Theorem

A continuous function on a closed interval must attains its extreme values somewhere in this interval.

Extreme Value Theorem

Extreme Value Theorem fails if no continuity or closed interval condition.

How to Find Global Max and Min

Global max = max of local maxima;Global min = min of local minima.

Find Local Max and Min

Understand Fermat’s Theorem

f(x) attains local max or min at c implies f’(c)=0, if f’(c) exists;reversely, f’(c)=0 does NOT necessarily imply local max or min.

f(x) might attain local max or min even when f’(c) does not exist.

Critical Point (Number)

Find Global Max and Min: Closed Interval Method

Examples