We talked about absolute maxima and minima. These are the tallest peaks and the lowest valleys in the entire mountain range. What about peaks of individual mountains and bottoms of individual valleys? The derivative, being a local concept, is like walking around in a fog; it cannot tell you if you are on the highest peak, but it can tell you whether you are at the top of some peak.
Let be a set and let be a function. The function is said to have a relative maximum at if there exists a such that for all where we have The definition of relative minimum is analogous.
For a differentiable function, a point where is called a critical point. When is not differentiable at some points, it is common to also say that is a critical point if does not exist. The theorem says that a relative minimum or maximum at an interior point of an interval must be a critical point. As you remember from calculus, one finds minima and maxima of a function by finding all the critical points together with the endpoints of the interval and simply checking at which of these points is the function biggest or smallest.
Suppose a function has the same value at both endpoints of an interval. Intuitively it ought to attain a minimum or a maximum in the interior of the interval, then at such a minimum or a maximum, the derivative should be zero. See Figure 4.4 for the geometric idea. This is the content of the so-called Rolle’s theorem 1
Named after the French mathematician Michel Rolle (1652–1719).
As is continuous on it attains an absolute minimum and an absolute maximum in We wish to apply Lemma 4.2.2, and so we need to find some where attains a minimum or a maximum. Write If there exists an such that then the absolute maximum is bigger than and hence occurs at some and therefore On the other hand, if there exists an such that then the absolute minimum occurs at some and so If there is no such that or then for all and then for all so any works.
For a geometric interpretation of the mean value theorem, see Figure 4.5. The idea is that the value is the slope of the line between the points and Then is the point such that that is, the tangent line at the point has the same slope as the line between and The name comes from the fact that the slope of the secant line is the mean value of the derivative, so the average derivative is achieved in the interior of the interval.
The theorem follows from Rolle’s theorem, by subtracting from the affine linear function with the derivative with the same values at and as That is, we subtract the function whose graph is the straight line and Then we are looking for a point where this new function has derivative zero.
The mean value theorem has the distinction of being one of the few theorems commonly cited in court. That is, when police measure the speed of cars by aircraft, or via cameras reading license plates, they measure the time the car takes to go between two points. The mean value theorem then says that the car must have somewhere attained the speed you get by dividing the difference in distance by the difference in time.
Let us look at a few applications of the mean value theorem. The applications show the typical use of the theorem, which is to get rid of a limit by finding the right sort of points where the derivative is not just close to some difference quotient, but actually equal to one. First, we solve our very first differential equation.
Now that we know what it means for the function to stay constant, we look at increasing and decreasing functions. We say is increasing (resp. strictly increasing) if implies (resp. ). We define decreasing and strictly decreasing in the same way by switching the inequalities for
The proof of i is left as an exercise. Then ii follows from i by considering instead. The converse of this proposition is not true. The function is strictly increasing, but
Another application of the mean value theorem is the following result about location of extrema, sometimes called the first derivative test. The result is stated for an absolute minimum and maximum. To apply it to find relative minima and maxima, restrict to an interval
We prove the first item and leave the second to the reader. Take and a sequence such that for all and By the preceding proposition, is decreasing on so for all As is continuous at we take the limit to get
Another often used application of the mean value theorem you have possibly seen in calculus is the following result on differentiability at the end points of an interval. The proof is Exercise 4.2.13.
The proof follows by subtracting and a linear function with derivative The new function reduces the problem to the case where That is, is increasing at and decreasing at so it must attain a maximum inside where the derivative is zero. See Figure 4.6.
The function is also differentiable on Compute Thus As the derivative is the limit of difference quotients and is positive, there must be some difference quotient that is positive. That is, there must exist an such that
or Thus cannot possibly have a maximum at Similarly, as we find an (a different ) such that or that thus cannot possibly have a maximum at Therefore, and Lemma 4.2.2 applies: As attains a maximum at we find and so
We have seen already that there exist discontinuous functions that have the intermediate value property. While it is hard to imagine at first, there also exist functions that are differentiable everywhere and the derivative is not continuous.
We claim that is differentiable everywhere, but is not continuous at the origin. Furthermore, has a minimum at but the derivative changes sign infinitely often near the origin. See Figure 4.7.
Figure4.7.A function with a discontinuous derivative. The function is on the left and is on the right. Notice that on the left graph.
And, of course, as tends to zero, tends to zero, and hence goes to zero. Therefore, is differentiable at 0 and the derivative at 0 is 0. A key point in the calculation above is that see also Exercises 4.1.11 and 4.1.12.
It is sometimes useful to assume the derivative of a differentiable function is continuous. If is differentiable and the derivative is continuous on then we say is continuously differentiable. It is common to write for the set of continuously differentiable functions on
Finish the proof of Proposition 4.2.8. That is, suppose is an interval and is a differentiable function such that for all Show that is strictly increasing.
Prove the following version of L’Hôpital’s rule. Suppose and are differentiable functions and Suppose that when and that the limit of as goes to exists. Show that
Prove the theorem Rolle actually proved in 1691: If is a polynomial, for some and there is no such that then there is at most one root of in that is at most one such that In other words, between any two consecutive roots of is at most one root of Hint: Suppose there are two roots and see what happens.