Relative and Absolute Extrema of a Function

Date: 01/07/2004 at 12:27:31
From: barbara
Subject: Relative and absolute extrema

What is the difference between the absolute extrema and the relative 
extrema in calculus?

Date: 01/07/2004 at 16:09:24
From: Doctor Peterson
Subject: Re: Relative and absolute extrema

Hi, Barbara.

A relative maximum is the greatest IN ITS NEIGHBORHOOD.  An absolute 
maximum is the greatest ANYWHERE (in the domain).

Suppose you wanted to find the highest point in your state.  Every 
mountain peak would be a relative maximum; it is a place where the 
slope is zero (or undefined), and every point nearby is lower.  The 
highest of those mountains MIGHT be the highest point or absolute
miximum in the state--but not necessarily.  In some states, the
boundary is on the slope of a mountain range whose peaks are in a
neighboring state; so the highest point might be on the slope of one
of those mountains.  This is in fact true of my home state,
Connecticut, where a nearby mountain peak within the state is almost
as high, but is not the absolute maximum:

    http://americasroof.com/ct.shtml 

Here is a picture of two relative maxima and the absolute maximum in 
such a case, with a finite domain:

    |                                            *
    |                               *            |
    |                            *  |  *      *  |
    |                           *   |   *  *     |
    |                               |            |
    |         *|*              *    |            |
    |     *    |    *               |            |
    | *        |      *       *     |            |
    *          |       *     *      |            |
    |          |         * *        |            |
    |          |                    |            |
   -+----------+--------------------+------------+
           relative             relative      absolute
              max                  max          max

To find the absolute maximum, you have to find all relative maxima, 
as well as boundary points, and determine which of these candidate 
maxima is the highest.

All the same things can be said about minima.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/08/2004 at 09:35:21
From: barbara
Subject: Thank you

Thank you very much for helping me with my math question.  Your answer
was very helpful, and I appreciate your time.