Relative and Absolute Extrema of a FunctionDate: 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. |
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