Section 4.4The Derivative in Graphing and Applications- “Absolute Maxima and Minima”

All graphics are attributed to:

•Calculus,10/E by Howard Anton, Irl Bivens, and Stephen DavisCopyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

Introduction

•In this section we will find the highest and lowest points over the entire “mountain range” instead of the high and low points in their immediate vicinity.

•In mathematical terms, we will be looking for the largest and smallest values of a function over an interval.

Absolute Extrema

•If a function has an absolute maximum at a given point in an interval, then the y-value associated with that point is the largest value of the function on the interval.

•Likewise for the absolute minimum and the smallest y-value of the function on the interval.

•There is no guarantee that a function will have an absolute max. or min. on a given interval.

•See examples on two following slides.

Absolute Extrema Examples

Absolute Extrema Examples con’t

The Extreme Value Theorem•The extreme value theorem tells us under

which conditions absolute extrema exist.•We will discuss how to find them on later

slides.

• In other words, if the function is continuous on [a,b], then the absolute extrema occur either at the endpoints of the interval or at the critical points inside (a,b).

Graphical Examples

Open Interval Application

•This is also valid on infinite open intervals such as .

How to Find the Absolute Extrema• This is very similar to finding the relative extrema. • After finding all of the critical points (derivative =

0, solve for x and non-differentiable points), find out which of them is the smallest and largest by substitution into f(x).

Polynomial Example

smallest

largest

Zero ProductProperty

Absolute Extrema on Infinite Intervals

• You must look at the end behavior of a function when determining whether or not a function has an absolute maximum or absolute minimum on

Infinite Interval Example

Absolute Extrema on Open Intervals

• A continuous function may or may not have absolute extrema on an open interval.

• There are certain conditions that will help determine whether or not they exist.

Examples

•See examples on pages 270-271

Tea at the Beverly Hills Hotel