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MATH 1121.81 Calculus I |
Chapter 3:
Applications of Derivatives
Further results of the Continuous Image Theorem from Topology:
Back in Chapter 1, I mentioned an important result from the topology of the real line that is usually stated:
The continuous image of a closed interval is either a closed interval or a single point.
The first important result we saw from that was the Intermediate Value Theorem.
The second result is even more important in applications. It is known as the Extreme Value Theorem (or the Max-Min Theorem, although there's another theorem, below, that we'll save the name "max-min" for in a Calculus class.)
One of the main applications of mathematics today is the area of problems referred to as "optimization problems," where we try to use mathematics to find an optimal solution to a problem in a particular situation. Many times "optimal" can be translated as "highest" or "lowest." For example, you may want to minimize cost of production, or maximize sales, or find the highest point reached by an object, or the slowest rate of growth for a disease, or....
In any of these cases, it is not just enough to know what the extreme value is, but also, how to achieve that value, or even, whether it is possible to achieve an extreme value.
The Extreme Value Theorem is what's known as an "existence theorem." It says that, under the right conditions, an extreme value exists, and some method of achieving that extreme value also exists. What it doesn't say is how to find that method or that value. But it's nice to know that something exists, before you start looking for it.
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Extreme Value Theorem: Let f(x) be a continuous, real-valued function defined on a closed interval, [a , b]. Then A) There exists a point, cM, in the interval [a , b], such that M = f(cM) >= f(x) for all x in [a , b]. B) There exists a point, cm, in the interval [a , b], such that m = f(cm) <= f(x) for all x in [a , b]. M is called the maximum value of f(x) on [a , b]. m is called the minimum value of f(x) on [a , b]. In terms of the Continuous Image Theorem, if the continuous image of [a , b] is a closed interval, then this interval is [m , M]. If the continuous image is a single point, then it is m = M. |
The Extreme Value Theorem says that there is a maximum (resp. minimum), and that there is at least one way of achieving that maximum (resp. minimum) within the interval [a , b].
Take a look at the GeoGebra applet below:
The next question becomes, how do we find these points, cm and cM?
The answer is found in the Max-Min Theorem for Derivatives, which provides a method of locating candidates for the extreme points. These candidates are then checked out, one after another, until the extreme values are found.
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The Max-Min Theorem for Derivatives: Let f(x) be continuous on the closed interval [a , b], with maximum value of M occurring at cM, and minimum value of m occurring at cm. Then cM and cm are in the set S, which consists of the following: A) The left-hand endpoint, a. B) The right-hand endpoint, b. C) All values x in (a , b) such that f ' (x) does not exist. D) All values x in (a , b) such that f ' (x) = 0. Points of type C) or D) are called "Critical Points" of the function f(x) in the interval (a , b). |
The proof of this theorem is important, instructive, beautiful and fairly simple. You should understand it, and really should memorize it. It boils down to assuming that A), B) and C) do not hold for a point, cm or cM, and then showing that D) must hold. That's the "Key Lemma" below:
Here are several examples of this theorem in use. What you do, each time, is:
First, make sure that the assumptions hold ( f(x) is continuous on [a , b].)
Second, set up a table that you fill with candidates from the set, S, starting with the end points.
Third, finish filling the table with the critical points (make sure that the critical points are in the interval!)
Finally, evaluate f (x) for all the points in the table (be sure to use f (x), and not f ' (x) ).
Automatically, the largest value for f(x) must be M, the maximum, and the smallest must be m, the minimum:
Next Example -- Be sure to look at places where f ' (x) is undefined, as well:
Final Example: More typical of what you'll do on the homework: Note the importance of being able to calculate exact solutions, especially when they come out "nice:"
Next Time, The Mean Value Theorem for Derivatives: