Taylor’s Theorem and Derivative Tests for Extrema



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Taylor’s Theorem and Derivative Tests for Extrema and Inflection Points

Sheldon P. Gordon

Department of Mathematics

Farmingdale State University of New York

gordonsp@farmingdale.edu
Abstract The standard derivative tests for extrema and inflection points from Calculus I can be revisited subsequently from the perspective of Taylor polynomial approximations to provide additional insights into those tests, as well as to extend them to additional criteria.
Keywords Taylor approximations, derivative tests for extrema, second derivative test, derivative tests for inflection points
Introduction Many of us tend to think of Taylor’s theorem as the capstone for the first year of calculus. But, if Taylor’s theorem is truly a capstone, then it should provide some broader perspectives on and deeper insights into topics that were previously encountered. Unfortunately, too few of us take the time to take advantage of some of these opportunities.

In a previous article [1], the present author illustrated how Taylor polynomials can be used to provide an understanding of why one gets the results of many of the limit problems, such as



,

that are typically encountered in calculus. Such limits are usually evaluated using l’Hopital’s rule, which gives one the correct answer, but provides little in the way of understanding why that answer arises. The use of Taylor polynomials provides the accompanying understanding.

In the present article, we describe how we may create another such opportunity to take a different view on a standard calculus topic by revisiting the derivative tests for extrema and for inflection points from the perspective of Taylor approximations. Throughout, we assume that the function under consideration has derivatives of all appropriate orders in an open interval centered at x = a.
Tests for Extrema The second derivative test for extrema from Calculus I states:

Suppose that a function f is such that f ‘(a) = 0. Then

(a) if f  (a) > 0, the function has a relative minimum at x = a

(b) if f  (a) < 0, the function has a relative maximum at x = a

(c) if f (a) = 0, the test is inconclusive.

All the introductory calculus textbooks treat the last case by showing a variety of examples in which the critical point turns out to be a maximum, a minimum, or neither to illustrate that anything can happen when f (a) = 0 depending on the sign and behavior of f ‘ near x = a. Some of the standard examples are the functions f(x) = x4, which has a minimum at the critical point x = 0, g(x) = -x4, which has a maximum at the critical point x = 0, and f(x) = x3, which has neither a maximum nor a minimum at the critical point x = 0, although it has an inflection point there.

However, Taylor approximations provide a tool for investigating precisely what is happening at such critical points and, more importantly, explain clearly why we get those results. Unfortunately, none of the standard calculus texts return to this issue after introducing Taylor series and Taylor approximations to make a connection between polynomial approximations and the derivative tests. Although these ideas are presented in advanced calculus courses and in texts such as [1], [2] or [3], only a very small number of the students from first year calculus ever take such upper division courses. As such, the overwhelming majority of calculus students lose the opportunity to see an interesting application of Taylor polynomials that provides fresh insights into a topic they have seen previously.

Let’s start by considering the quadratic Taylor polynomial for a function f centered at x = a:



.

If f has a critical point at x = a where f’(a) = 0, this approximation reduces to



.

From this, one can conclude that whenever f (a) > 0, near x = a the function behaves like a parabola with vertex at (a, f(a)) and a positive leading coefficient. Therefore, the critical point at x = a is the location of a relative minimum. Similarly, the function must have a relative maximum at x = a whenever f (a) < 0. Finally, when f(a) = 0, it is clear that the test is inconclusive.

However, additional insight into this situation can be provided by looking at higher degree Taylor polynomials. The cubic Taylor polynomial approximation for f centered at x = a is

.

If f (a) = 0 and f  (a) = 0, then the Taylor approximation reduces to



,

so that, near x = a, the behavior of f(x) is similar to that of the cubic function C(x – a)3, where C is a real number. From this, we immediately have the following result:


Third Derivative Test for Extrema Suppose that a function f has a critical point at x = a such that f (a) = 0 and f (a) = 0. Then

(a) if f (a)  0, the function cannot have an extremum at x = a

(b) if f  (a)  0, the function must have an inflection point at x = a

(c ) if f  (a) = 0, the test is inconclusive.


Unfortunately, it may appear that we may have traded in one inconclusive situation with the second derivative test for another inconclusive situation with the third derivative test. But Taylor’s theorem lets us pursue this issue as far as we like. Indeed, suppose that a function f has a critical point at x = a where f, f, and f are all zero. The fourth degree Taylor approximation for f about x = a then reduces to

.

Therefore, near x = a, the behavior of f(x) is similar to that of the quartic function C(x – a)4, where C is a real number. We therefore have the following result.


Fourth Derivative Test for Extrema Suppose that a function f has a critical point at x = a such that f (a) = 0, f (a) = 0, and f (a) = 0. Then

(a) if f (a) > 0, the function has a minimum at x = a

(b) if f (a) < 0, the function has a maximum at x = a

(c ) if f (a) = 0, the test is inconclusive.


Clearly, we could continue this process indefinitely. Formal theorems that summarize these results or problems that ask the students to devise such tests, can be found in the references [2] – [5]. For example, Kaplan [4] states:

Let f ‘(x0) = 0, f “ (x0) = 0, …, f (n)( x0) = 0, but f (n+1)( x0)  0; then f(x) has a relative maximum at x0 if n is odd and f (n+1)( x0) <0; f(x) has a relative minimum at x0 if n is odd and f (n+1)( x0) > 0; f(x) has neither relative maximum nor relative minimum at x0 but a horizontal inflection point at x0 if n is even.

Similarly, McShane and Botts [5] propose the following as a problem:

Let c be an interior point of the interval [a, b] where a function f is defined and continuous together with its first n derivatives. In order for f to have a relative minimum at c:

(a) it is necessary that f ’(c) = f (c) = … = f (n) (c) = 0 or else that the first of the derivatives f’(c), f”(c) …, f (n) (c) which is not zero be of even order and positive;

(b) it is sufficient that there be a positive even integer m such that m < n and f ’(c) = f (c) = … = f (m – 1)(c) = 0 and f (m)(c) > 0.


Criteria for Inflection Points

The notion of inflection point has become considerably more prominent in the so-called calculus reform texts and even non-calculus-based uses of the idea enters into the discussions in some of the modern college algebra and precalculus texts. Consequently, let’s consider criteria for inflection points. The usual test applied in calculus is that a function f has an inflection point at x = a where either f (a) = 0 or f (a) is undefined and f  changes sign about the point x = a. At various times, different students of mine in calculus have presumed the existence of a third derivative test to conclude the existence of an inflection point. We now investigate this possibility using a slightly different application of Taylor approximations.

In particular, suppose we write the quadratic Taylor approximation for the first derivative f (x) about x = a:

.

Suppose that f (a) = 0, so that this reduces to



.

Therefore, if f (a) > 0, then f  behaves like a quadratic with a positive leading coefficient and so has a relative minimum at x = a. However, if f ’ has a minimum at x = a, then the derivative is decreasing to the left of x = a and is increasing to the right of the point and consequently f has an inflection point at x = a. The parallel argument applies if f (a) < 0. We therefore see that there is indeed such a third derivative test for inflection points.

Alternatively, we could “integrate” the above Taylor approximation for f  and obtain a cubic approximation to f and come to the same conclusion. Either way, we have the following analog of the second derivative test for extrema:
Third Derivative Test for Inflection Points Suppose that a function f is such that f (a) = 0 and f (a) = 0. Then

(a) if f (a)  0, the function has an inflection point at x = a

(b) if f (a) = 0, the test is inconclusive.
The interested reader can investigate the inconclusive result in part (b) by looking at the cubic and higher degree Taylor approximations to f , but we will not do so here.

Some Examples


We now illustrate the above results with some specific examples. First, consider the function whose critical points occur when = 0, so that x = 0. Notice that

,

which is zero when x = 0. Consequently, the second derivative test is inconclusive. Also,



,

and this is equal to -6 when x = 0. So, by the Third Derivative Test for Extrema, the function cannot have an extremum at x = 0. However, by the Third Derivative Test for Inflection Points, we conclude that the function must have an inflection point at x = 0.

Since we are using Taylor approximations to provide insight into the behavior of functions with these conditions, it makes sense to look at the specific Taylor expansions of the specific function. The corresponding Taylor polynomial approximation is

and we therefore see why the function must have an inflection point at x = 0. We show the graphs of both the function (marked in dashes) and the Taylor approximation (as a solid curve) in Figure 1, from which it is also evident that there must be an inflection point there.

As a second example, consider the function whose critical points occur when = 0, so that x = 0. Notice that

and ,

both of which are zero when x = 0. However,



,

which is equal to -24 when x = 0. Consequently, using the Fourth Derivative Test for Extrema, we conclude that there is a relative maximum at x = 0. This can be seen clearly when we consider the Taylor approximation  1 - x4, as well as in Figure 2 where we show both the function (dashed) and the Taylor polynomial (solid).

Finally, as a third example, consider the function f(x) = x3 cos x. We have

f ‘(x) = 3x2 cos x - x3 sin x,

which is zero when x = 0. (There are infinitely many other critical points that occur whenever x tan x = 3, but we will not consider any of them here.) Notice that



f “(x) = 6x cos x – 6x2 sin x - x3 cos x,

so that f “(0) = 0. Moreover,



f ‘“(x) = 6 cos x – 18x sin x – 9x2 cos x + x3 sin x,

and f ‘”(0) = 6. Therefore, using the Third Derivative Test for Extrema or the Third Derivate Test for Inflection Points, we conclude that there is an inflection point at x = 0. This is evident from the Taylor approximation



f(x) = x3 cos x x3 [x – x3/6] = x3x5/6,

as well as from Figure 3, which also shows the original function (dashed) versus the Taylor polynomial (solid).


Pedagogical Implications The ideas in this article can be used for a short classroom discussion when Taylor approximations are being covered at the end of Calculus II. These ideas provide a nice application of the concepts at the same time giving a fresh perspective on the methods for identifying and testing extrema from Calculus I.

Alternatively, an instructor can construct a series of guided home work problems asking the students to explore these ideas on Taylor approximations while simultaneously having the students review the derivative tests from Calculus I.


References

1. Gordon, Sheldon P, l'Hopital's Rule and Taylor Polynomials, Int'l J of Math in Sci & Engg Ed, 1992.

2. Apostol, Tom, Mathematical Analysis: A Modern Approach to Advanced Calculus, 2nd Ed., Addison-Wesley, 1974.

3. Friedman, Avner , Advanced Calculus, International Thompson Publ., 1971.

4. Kaplan, Wilfred, Advanced Calculus, 5th Ed., Addison-Wesley, 2002.

5. McShane, Edward James, and Truman Arthur Botts, Real Analysis, Van Nostrand Publ, 1959.


Acknowledgement The work described in this article was supported by the Division of Undergraduate Education of the National Science Foundation under grants DUE-0089400 and DUE-0310123. However, the views expressed are not necessarily those of the Foundation.
Biographical Sketch Sheldon Gordon is Professor of Mathematics at SUNY Farmingdale. He is a member of a number of national committees involved in undergraduate mathematics education and is leading a national initiative to refocus the courses below calculus. He is the principal author of Functioning in the Real World and a co-author of the texts developed under the Harvard Calculus Consortium.

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