Summary

Usually we want to know what number of units to produce in order to maximize profit - not just what number gives a local maximum! Your boss doesn't want to hear you say ``compared to seven and nine units, producing eight units loses us less money!'' if, in fact, 4000 units produces a profit of a million dollars.... That's an example of the difference between relative and absolute extrema.

This is the big difference between relative and absolute extrema: the absolute maxima are the kings of the hill, and the absolute minima are the basement of the building. They're the highest, or the lowest: no one's got them beat (although there may be ties, which accounts for the need to talk about them in plural!).

Absolute extremum: Let c be a number in the domain of a function f. Then f(c) is an absolute (or global) maximum for f if for all x in the domain;, and f(c) is an absolute (or global) minimum for f if for all x in the domain.

First of all, we should note that an absolute extremum is always a relative extremum (but not vice versa generally).

Extreme Value Theorem: A function f that is continuous on a closed interval [a,b] will have both an absolute maximum and an absolute minimum on the interval.

This is another theorem (like the intermediate value theorem) that is so obvious and yet perhaps mystifies you: it says that, should you take a walk, at some point in your walk you attain your highest elevation and at some point in your walk you attain your lowest elevation. Try to imagine violating this rule!

Finding Absolute Extrema: To find absolute extrema for a function f continuous on a closed interval [a,b]:

1. Find all relative extrema, by finding all critical numbers and evaluating f at all of them.
2. Evaluate f for the endpoints a and b of the interval [a,b]. Don't forget the endpoints!
3. The largest value found is the absolute maximum, and the smallest is the absolute minimum for f on [a,b].

Examples:

1. Problems #1-8, p. 332