Mat210 Section 3.4 The Extreme Value Theorem

Now that we have your attention, we make the observation that in today's "reality show" TV culture, we see the "extreme" of different situations being explored and exploited as entertainment. Naturally the "extreme" suggests the most of something or the least of something, and that is exactly what we mean when we use the term in the mathematical sense. An absolute extremum (Latin) is the greatest y value on some interval, or the smallest y value on some interval. We have already studied how to find the relative extrema of a function, that is, the relative maximum and minimum values, using the first derivative. What more must be done to go from a relative extremum to an absolute extremum? This is the point of this lesson that follows.

### Finding the Absolute Extrema

The Extreme Value Theorem tells us that we can in fact find an extreme value provided that a function is continuous. Thus, before we set off to find an absolute extremum on some interval, make sure that the function is continuous on that interval, otherwise we may be hunting for something that does not exist. Here is the procedure for finding the absolute extreme value(s) of a function on an interval.
Finding Absolute Extrema of f(x) on [a,b]:
1.  Find all critical points of f(x) that are inside (in the interior) of the interval [a,b].  This makes sense if you think about it.  Since we are only interested in what the function is doing in this interval we don't  care about critical points that fall outside the interval.
2.  Evaluate the function at the critical points, that is, where y ' = 0 or where the derivative fails to exist. (Sharp points).
3.  Evaluate the function at the end points of the interval.  That is, find f(b) and f(a).
4.  Identify the largest and smallest y values found in that interval. Those are the absolute extrema.
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Let's work through some examples.

Example 1: Consider the three curves shown below. State whether the absolute maximum / minimum values occur on the interior of the interval [a, b] or at the endpoints. Roll your mouse over the Extreme Value Theorem to check your answers.

Example 2: Locate the value(s) where the function attains an absolute maximum and the value(s) where the function attains an absolute minimum, if they exist, on the given interval. . Graph the function to verify your conclusions.

1. Since f(x) is continuous, it will attain a maximum and a minimum value somewhere on the interval from x = -3 to x = 3 (so says the theorem.) Since the function is a polynomial, there won't be any sharp peaks or discontinuities to be concerned about.

2. Find the derivative, set it equal to zero, and solve for x. Then find the y value at that x. What do you get?

3. Check the endpoints. This means, find f(-3) and f(3). What y values do you get at the endpoints?

4. Compare your results from steps 2 and 3. What is the larges value, and what is the smallest value?

5.  A quick check on the graphing calculator gives a visual verification that our results make sense. The endpoints will have smaller/larger values than the interior relative max and min.,

Example 3: In our checklist at the top of the page, we are reminded to check places where the derivative fails to exist. It is important to examine a case where this crops up.

 Problem: Find the absolute maximum and the absolute minimum values of the given function on the closed interval [ -2, 3].  Graph the following function on your graphing calculator: using the following window dimensions: [- 5, 5] X [- 5, 5] . You should get a screen similar to the following (remember to put parentheses around the fractional exponent.)

Example 4:
The profit for a company Extreme Widgets Inc. is given by:

where x is the number of widgets made and sold (in thousands) and P is the profit in thousands of dollars. However, the maximum capacity the plant has currently to produce widgets is 15 thousand widgets . Find where the profit attains an absolute minimum and an absolute maximum value. Graph the function and visually confirm you results.

1. Since f(x) is continuous, it will attain a maximum and a minimum value somewhere on the interval from x = 0 to x = 15. Since the function is a polynomial, there won't be any sharp peaks or discontinuities to be concerned about.

2. Find the derivative, set it equal to zero, and solve for x. Then find the y value at that x. What do you get?

3. Check the endpoints. This means, find f(0) and f(15). What y values do you get at the endpoints?

4. Compare your results from steps 2 and 3. What is the larges value, and what is the smallest value?

5. Check the graph. Are your results reasonable?