Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques o dierentiation and the graphical meaning o a derivative. What you can learn here: How to use derivatives to identiy the extreme values, that is maxima and minima, o the graph o a unction. The derivative was designed to provide the slope o the graph o a unction, so it is not surprising that it can also provide urther useul inormation about such graph. It is now time to make ull use o this connection and to develop ways to analyze and identiy eatures o the graph o a unction by using derivatives. As it is so oten the case in mathematics, it will be useul to begin by ollowing Aristotle s advice and deine some concepts precisely, or at least as precisely as we need them to be. Deinition An absolute minimum or a unction x is a point c, ( c ) on the graph o x that is not higher than any other point o such graph. An absolute maximum or a unction x is a point c, ( c ) on the graph o x that is not lower than any other point o such graph. In simple words, an absolute maximum is the highest point and an absolute minimum is the lowest point on the whole graph, except possibly or other points with the same y-coordinate. Beore anyone starts objecting to these deinitions, let me state that many authors reer to the number c as the maximum or minimum o a unction, not Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 1 the point. Such a choice makes sense rom many points o view and I respect it highly. However, or a number o reasons related to my desire to explain these concepts in a simple way, I preer to give the name o maximum or minimum to the point, not just its y coordinate.
The same applies to the next deinitions. I the point minimum o or extremum or Deinition x. c, c is an absolute maximum or x, then we call it an extreme point In mathematics, whenever an object is deined we ask three basic questions: 1) Under which conditions can we be sure that it exists? (Existence) ) Is there only one such object in any given situation, or can there be several? (Uniqueness) 3) How do we ind one? (Construction) In the case o extreme points, it is clear, by considering the sine and cosine unctions, that they are not unique, so do not be surprised i you ind more than one extreme point or the same unction. What about the other two questions? The existence question has a nice answer in the ollowing classic theorem. Technical act The Extreme Value Theorem I a unction x is continuous on a closed interval ab,, then it has at least one absolute maximum and at least one absolute minimum on ab., The ormal proo o this theorem is rather technical and I will skip it. I you are interested in seeing one, you can start here. However, it is an important theorem since it can be used both in later calculus theory and in applied problems. 3 y x x x 3, 1 3 This is a polynomial unction, so it is continuous. Since we are restricting its domain to the closed interval 1, 3, the extreme value theorem assures us that it has at least one absolute maximum and one absolute minimum. The graph, shown here, conirms this conclusion, although we are still not sure o where these points occur. It looks like they occur at x 1, 0,, 3, but we cannot be sure based on the graph only. We need to wait until we see the answer to the construction problem. 3 y x x x 3, 1 3 This is the same unction as beore and its graph looks exactly the same on the calculator, BUT But the end points o the domain are NOT included and thereore the Extreme Value Theorem does not apply and does not guarantee the existence o extreme values. Notice, however, that this unction still has an absolute maximum, at or near x 0, and an absolute minimum, at or near x. The act that we cannot use the theorem does not mean that the unction has no absolute extrema: it is a oneway only theorem! This last example shows a weakness o existence theorems: they do not tell us how to ind what we are looking or, nor can they assure us o the absence o the Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page
eature. So we need to look at how to solve the construction problem. Beore doing that, we need a ew more deinitions. A relative or local maximum or a unction c, ( c ) x is a point on the graph o Deinition x that is not lower than any other point o such graph on some x interval ab, around c. A relative or local minimum or a unction c, ( c ) x is a point on the graph o x that is not higher than any other point o such graph on some x interval ab, around c. Local maxima and minima are collectively called local extrema. I agree, and I hope you have seen the next two as well: I include all these deinitions or your convenience and or completeness. By the way, you will ind that the next two deinitions are also presented a little dierently by some authors. Again, my choice is based on my desire to keep things simple. A cut point o a unction such that either: c 0 or c domain o Deinition y x is a value x c DNE and c is at the boundary o the x x x x 4 1 x, because the unction becomes 0 x, because the unction is undeined there AND these are at The cut points o this unction are there, and 1 the boundary o the domain o the unction, since such domain is:, 11, D Notice that the unction is also undeined at 0 x, or anywhere between -1 and 1, but these are not cut points as they are totally outside the domain. In simple words, a relative or local extremum is the highest or lowest point or a section o the graph that does not include end points Lots o deinitions! I am glad that I have seen them beore! Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 3
Deinition x that is in the domain o A critical value o a unction x x 4 x 1 The derivative o this unction is: x x is a cut point o x. x x 1 x 4 x x4 x1 x1 with cut points at x 1 5 (derivative becomes 0) and x 1 (derivative is undeined). However, only x 1 5 are critical values, since they are the only ones in the domain o And now, drum roll please! x. Technical act The extended Fermat s theorem I a unction c, c, then c is a critical value o x has a relative extremum at x. Proo Again the proo is rather technical and not useul or our goals and you can ind it here. However, to convince yoursel o its truth, just consider the act that i c, the unction s slope is positive, hence the unction is going up, 0 while i c 0, the slope is negative and the unction is going down. But at a relative extremum the unction is neither going up nor down! Thereore, we are let with the other two options: either x 0 or it is undeined. Keep in mind that this is not a ormal proo, just a simple argument that I hope can convince you. x x 4 x 1 We just saw that this unction has critical values at x 1 5. I we look at the graph o this unction, we notice that at x 1 5 there is a relative maximum and at x 1 5 a relative minimum. Notice that neither is an absolute extremum, but that no other relative extrema exist or this unction. What does the other cut point o the unction correspond to? Why did you call it the extended Fermat s theorem? Because Fermat only looked at the case o dierentiable unctions, thus excluding the case when the derivative is undeined. But that case its nicely into the statement o the theorem, so I included it, again or simplicity. This theorem also gives us a great strategy to look or extreme points. Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 4
Strategy or inding extreme points o a unction To identiy all extreme points o a unction x : 1. Determine all critical values o x, including the end points o its domain.. Check whether each o them is a relative minimum, a relative maximum, or neither. 3. Among all relative extrema, choose the absolute ones by picking the highest and lowest points. What do you mean by neither? Doesn t it have to be one or the other? No! The theorem states that i there is an extremum, it occurs at a critical value, but it does not claim that every critical value corresponds to an extremum. Some critical values are neither! y 3 x 5 The derivative o this unction is x 0 y 3x, so that its only critical value is at. But as we can see rom the graph, the unction has neither a maximum nor a minimum there. /3 y x x x, 1 1 This unction is continuous and its domain is a closed interval, so, by the EVT, it has both absolute maximum and minimum. But where are they, and does the unction have other relative extrema? Here is a irst graph: 3 y x x x 3, 1 3 The derivative o this unction is: y x x x x x 3 6 3, 1 3 Notice that the derivative is not deined at the end points, since the unction is only continuous on one side. Thereore, the critical values are at x 0,, where the derivative is 0, and at x 1, 3 where the derivative is undeined, but the unction exists. These are exactly the values we ound earlier by looking at the graph. It looks like the end points are the only extrema, but let us use Fermat s theorem to conirm this. The derivative o this unction is: 1/ 3 1/ 3 1 3x y' x 1/ 3 3 3x This is 0 at 1 x and undeined at x 0 and at the end points x 1. 7 Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 5
These are all critical values, since they are all in the domain o the original unction. Thereore, any extrema must be at some o these points, but which ones and are they all extrema? By looking at the previous graph, we can see that there is an absolute maximum at x 1 and an absolute minimum at x 1, but what about the other two? By using a more ocused window we can see them more clearly: Do we have to rely always on the calculator s graph to check what kind o extreme point we have at a critical value? Certainly not, since the irst developers o calculus did not have a calculator and still could igure it out! I suspect that you either know or have a pretty good idea o how to classiy critical values, that is, how to decide whether they provide a maximum, minimum or neither. But we ll discuss a proper strategy to do that in the next section. We have a relative minimum at x 0, a relative maximum at 1 x. 7 Notice that in this example the relative minimum occurs or a value o x or which the derivative is undeined, NOT equal to 0. Summary An extreme point is one where the unction reaches a maximum or a minimum, either locally or globally. The Extreme Value Theorem tells us that every unction that is continuous over a closed interval reaches at least one absolute maximum and at least one absolute minimum. Fermat s theorem, in its extended version, tells us that relative extrema are only ound at the critical values, that is, at values in the domain o the unction where the derivative is either 0 or undeined. This generates a simple strategy to identiy all extreme points o a unction. Common errors to avoid Do not assume that every critical value corresponds to a maximum or a minimum: it may be neither! Don t ignore the end points o the domain, i they are included in the domain, since they are critical values too. Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 6
Learning questions or Section D 8-1 Review questions: 1. Explain what the Extreme Value Theorem states.. Explain what Fermat s Theorem states. 3. Describe the dierence between the Extreme Value Theorem and Fermat s Theorem, but don t just restate them! 4. Describe the strategy or identiying all possible extreme points o a unction. Memory questions: 1. Which theorem guarantees the existence o extreme points?. Which theorem is used to ind the location o the extreme points? 3. What is the derivative o a unction at one o its maximum or minimum values? 4. What are the two conditions under which we can use the Extreme Value Theorem? 5. Which two conditions deine a critical value? Computation questions: For each o the unctions in questions 1-0, identiy and classiy all extreme values, that is: a) Use the Extreme Value Theorem to determine i extreme points exist. b) Use Fermat s Theorem to identiy which points can possibly be maximum or minimum points. c) Use the calculator s graph to determine i each o them is a maximum, a minimum or neither. y 1 x on, 1.. 1. 3 3 y x x 3 1 Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 7 3. y x x x 3 1 18 30
4. y 6x x 3 17. y x x e on [-3, 1] 5. 1 4 3 y 4x 3x 1x 10 18. y e e x x on [-1, 1]. 6. y x 1x 4 7. 1 y on 1 x 3. x 8. 1 y on 1 x 1. x 5x 9. y x 3 on (-1, 1) 19. y x ln x on [1, 4]. 3 1 0. y x ln x on,. 1. The unction 3 x x i x y x 4 1 9 x 1 has the graph shown here. 5x 10. y x 3 on [-1, 1] 5x 11. y x 3 5x 1. y x 3 on 0, 13. y x sin x on [-3, 3] 14. g( x) x cos x on 0, cos x sin x 15. x on 0, 16. y x tan x Use calculus to identiy all possible critical values o the unction and use the graph to classiy each such point as a maximum, minimum or neither.. For what values o a and b, i any, does the unction y minimum at 1,? b x 1 bx ax e have a ax 3. The unction x x e has a local minimum at can a and b be? 0,. What Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 8
3 4. A unction has ormula x ax bx x and has critical values at x 1,. Determine the values o a and b and classiy these two critical values. Theory questions: 1. I (3)=0, does that mean that at x=3 there is a maximum or a minimum?. Is it possible that a unction y x derivative is not 0? has a maximum at a point where the 3. Which values are cut points or a derivative, but are not critical values? 4. Any cubic unction is continuous, but has no absolute maximum. Why doesn t that contradict the Extreme Value Theorem? 5. Is it possible or a unction to have several critical values, but no maxima or minima? 6. How do we distinguish between extreme points and vertical asymptotes? 7. Why is the restriction o a unction to a closed interval useul in applications? a) Does the unction show the presence o local maximum and minimum points? b) Does the Extreme Value Theorem apply to this unction? c) Where do its relative maxima and minima seem to occur? d) What can we say about the derivative at each relative maximum or minimum? e) Does Fermat s theorem apply to this unction? 8. Use the graph shown here to answer the ollowing questions about the unction it represents: Proo questions: 1. Decide which unctions in the amily described by the ormula y x x c, c 0 have extreme points, i any, and i so, o what type. Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 9
Templated questions: 1. Identiy and classiy the extreme points o the unctions listed in the document Sample unctions to analyze.. Construct a reasonably simple unction and use the methods o this section to identiy and classiy its extreme points. What questions do you have or your instructor? Dierential Calculus Chapter 8: Graphical analysis Section 1: Extreme points Page 10