The presentation here, as in the textbook, jumps into a different mode. The idea here is not just to describe, measure, and compute derivatives, but to use those ideas and derive consequences of them. This is a brief sojourn into theoretical mathematics.
One of the primary applications of calculus is to optimization problems, finding how to configure a situation to result in the largest, or smallest, value of a particular output. Such things as finding the price of a product you are selling, to maximize the profit, or how to design a box to minimize the materials required to build it, are standard examples of optimization.
From a theoretical point of view, this theorem shows that an optimization problem will always have a solution under the right circumstances. While this is enough for a mathematician, it does not give any help to find the point where the optimum value occurs.
Theorem 1 If f is continuous on a closed interval [a,b] , then there is a point x0 Î [a,b] where f is largest ( f(x) £ f(x0) for all x Î [a,b] ), and a point x1 Î [a,b] where f is smallest ( f(x) ³ f(x0) for all x Î [a,b] ).
The importance of this result, which we actually don't prove, is that it asserts the existance of such a point. To be able to say that an object exists, with whatever property we're talking about, is very powerful in mathematics. This may seem totally obvious, but it actually isn't. In order to see that this is non-trivial, consider the following function:
This behavior is ruled out for continuous functions, and is one fact that makes continuous functions ``nice''.
Definition 1 If f(x) is defined on an interval [a,b] , a point x0 where f(x) has its largest value over [a,b] is called an absolute maximum point, and a point x1 where f(x) has its smallest value over [a,b] is called an absolute minimum point. These are sometimes called extreme points, or extrema. The values f(x0) and f(x1) are the maximum value of f(x) and the minimum value of f(x) , and they are referred to collectively as the extreme values of f(x) .
Definition 2 A point x0 where f(x) has its largest value over some open interval I containing x0 is called a local maximum point, and a point x1 where f(x) has its smallest value over some open interval I containing x1 is called a local minimum point. The values f(x0) and f(x1) are a local maximum value of f(x) and a local minimum value of f(x) , and they are referred to collectively as the local extreme values of f(x) . the points where a local maximum or local minimum happend are called local extreme points.
Often the word ``local'' is left out, even when that is really what is meant. Other authors and professors will call these ``relative'' extrema.
Where, at what point, does a maximum or minimum of f(x) occur? That is not, in general, an easy question to answer. But, there is a way to narrow down the search. Certainly the extreme points could be at the endpoints of the interval [a,b] . So we should always look at f(a) and f(b) . But where else? The answer is obvious if we graph the function; the maximum occurs at the top of a hill. It occurs at the height where no part of the graph goes above that height. This is not a big revelation, but, if the function is differentiable at that maximum point, since that maximum-height line is never crossed, the tangent line can't cross over the line either, so the tangent line is horizontal. This observation is what is sometimes called Fermat's Theorem. It has nothing to do with Fermat's Last Theorem; unlike that one, he may very well have proved this one.
Theorem 2 If f(x) has a local extremum at c , and if f¢(c) exits, then f¢(c) = 0 .
Remark 1 f¢(c) may not exist at all. The function |x| has a minimum over all x Î [-1,1] at the point 0, even though there is no derivative there. You have to be careful with this statement. It doesn't say that any point x with f¢(x) = 0 is a maximum or a minimum. It could be something like f(x) = x3 at x = 0 , which is not an extreme point. But, if it is an extreme point, the derivative will be 0 if it exists. We then only have to look at all the places where f¢(x) = 0 , or where f¢(x) doesn't exist (kinks in the graph), and the endpoints, to find where the function has its maximum or its minimum.
Proof.
Definition 3 A point where:
is called a critical point of f(x) . The values of f(x) at a critical point are called critical values, or critical numbers.
Example 1 Find the local maximum and minimum points and values, and the absolute maximum minimum points and values, of the function f(x) = 2sin(x)+cos(2x), x Î [0,2p]. Solution
This example illustrates a basic observation about critical points, usually called the first derivative test, since it tests whether a critical point is a local maximum or minimum by examining the first derivatives. From Corollary 3 of the next section, the proof of this result is easy.
Theorem 3 [First Derivative Test]. If x0 is a (non-endpoint) critical point of a function f(x) , and: 1) f¢(x) > 0 for x < x0 (but x near x0 ), and f¢(x) < 0 for x > x0 , then x0 is a local maximum point of f(x), 1) f¢(x) < 0 for x < x0 (but x near x0 ), and f¢(x) > 0 for x > x0 , then x0 is a local maximum point of f(x) .
1) f¢(x) > 0 for x < x0 (but x near x0 ), and f¢(x) < 0 for x > x0 , then x0 is a local maximum point of f(x),
1) f¢(x) < 0 for x < x0 (but x near x0 ), and f¢(x) > 0 for x > x0 , then x0 is a local maximum point of f(x) .
I think of this as ``increasing up to a maximum, decreasing beyond it'', or ``decreasing down to a minimum, and increasing away''. It really is useful to find the signs of the derivative and lay them out on a number line as in the previous example,
There is another derivative test, called the second derivative test. It's not quite so good as this one, though, for two reasons. The first is that you have to assume that f¢(x0) exists (note that the first derivative test does not mention that), and the second is that there is an indeterminate case.
Theorem 4 [Second Derivative Test]. If f¢(x0) = 0, and 1) f¢¢(x0) > 0 , then f(x) has a local minimum at x0 , 2) f¢¢(x0) < 0 , then f(x) has a local maximum at x0 . But, if f¢¢(x0) = 0 , it could be either a local maximum, or maybe a local minimum, or maybe neither one.
1) f¢¢(x0) > 0 , then f(x) has a local minimum at x0 ,
2) f¢¢(x0) < 0 , then f(x) has a local maximum at x0 . But, if f¢¢(x0) = 0 , it could be either a local maximum, or maybe a local minimum, or maybe neither one.
Exercise 1 Find the local and absolute maximum and minimum points, and their values, of the function f(x) = x4-2x3+x2+1, for x Î [-1,2] .
Answer:
Some of the results we spend a lot of time proving may seem trivial. If so, that is because they make explicit some standard assumptions you make about how functions behave. But functions only really behave that way if they are sufficiently nice, and most functions (in a very real sense, most of them) are really badly behaved. Theoretically, these results are not really difficult, but they are serious enough to be proved carefully.
This first theorem is always called Rolle's theorem, named after Michel Rolle who stated and proved this result in 1691.1
Theorem 5 If f is continuous on a closed interval [a,b] , and differentiable on the open interval (a,b) , and if f(a) = f(b) , then there is some point c Î (a,b) so that f¢(c) = 0 .
The only purpose that theorem has in life is to prove this next theorem, which we will use to verify a lot of facts you would have assumed, but which need proof.
Theorem 6. [Mean Value Theorem] If f is continuous on a closed interval [a,b] , and differentiable on the open interval (a,b) , then there is some point c Î (a,b) so that f¢(c) = f(b)-f(a)b-a . Proof.
Now, the Mean Value Thorem is often illustrated with the sort of example that shows the statement, but not the meaning, of the theorem. In a sense, what the statement of the theorem is, is that the slope of the curve at some point is the same as (f(b)-f(a))/(b-a) .
Here are two examples, one illustrating an unfortunately realistic application of the statement, but not the idea, and the other really getting at the idea of the MVT.
Example 2 My commute to work involves a stretch of the Northeast Extension of the Pennsylvania Turnpike. I get on (at Plymouth Meeting), mile marker 20, and am handed a ticket, with the time stamped on it. When I get off, at the Quakertown exit, mile 44, 21 minutes later, the ticket taker hands me my change and a speeding citation. Have I been wronged?
Solution
True enough, this is an application of the MVT. But the real import of this theorem is that it provides a statement about the existence of certain values of the derivative. Here is a better example.
Example 3 Show that the function f(x) = x5+4x3+3x-8 has only one root, one point x where f(x) = 0 .
Exercise 2 Show that the function f(x) = x6+5x4+2x2-7x+3 has at most two distinct roots.
Hint: Look at the roots of f¢(x).
1I have been teaching calculus roughly since then, and have only now learned the guy's first name, thanks to the internet.
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Copyright (c) 2000 by David L. Johnson.