6.2 Important Theorems

Transcription

1 6.2. IMPORTANT THEOREMS Important Theorems Local Extrema and Fermat s Theorem Definition (local extrema) Let f : I R with c I. 1. f has a local maximum at c if there is a neighborhood U of c such that f (x) f (c) for all x U I. 2. f has a local minimum at c if there is a neighborhood U of c such that f (x) f (c) for all x U I. 3. A local extremum is either a local maximum or a local minimum. Finding local extrema is made easier by the following theorem. Theorem Suppose that f has a local extremum at an interior point c of I. If f is differentiable at c, then f (c) = 0. Proof. We give a hint for the proof in the case f has a local maximum at c and ask the reader to complete the proof as part of the homework. Write the definition of f having a local maximum at an interior point c of I. Study the sign of f (x) f (c) x c on each side of c. Use the previous step to deduce the sign of the one-sided derivatives. Use the fact that f is differentiable at c, to finish the proof. The above theorem tells us exactly which numbers to look for when we are trying to find extreme values. If c corresponds to an extreme value, then either f (c) exists or it does not exist. If it exists, the above theorem tells us it must be 0. So, we conclude: Corollary A function f has an extreme value at an interior point c if either f (c) = 0 or f (c) does not exist. Definition (critical number) A number c which satisfies the conditions of the corollary is called a critical number. As you will remember from differential calculus, not every critical number corresponds to an extreme value. This is why when looking for extreme values, we first find critical numbers. Then, these critical numbers must be tested to see which ones correspond to a local maximum, local minimum or neither. In the homework, you will be asked to develop a procedure to find local extrema.

2 224 CHAPTER 6. DIFFERENTIATION Intermediate Value Property of the Derivative Earlier, we saw that being differentiable implied being continuous, the converse being false. The derivative itself needs not be continuous. We saw an example of a function continuous and differentiable at 0, whose derivative was not continuous at 0 (see problem 9). However, it turns out that even if the derivative is not continuous, it still has one of the very important properties continuous functions have: the intermediate value property. Theorem Let f : I R be a function. If f is differentiable on I, then f satisfies the intermediate value property on I. That is, given a, b in I with C between f (a) and f (b), f (a) f (b), then there exists c between a and b such that f (c) = C. Proof. We outline the proof and ask the reader to fill in the details as part of the homework. Without loss of generality, we may assume that a < b. Then, either f (a) < f (b) or f (a) > f (b) (since f (a) f (b)). We do the proof in the case f (a) < f (b), the other case is also part of the homework. In this case, f (a) < C < f (b). Define F : [a, b] R x f (x) Cx 1. Explain why F is differentiable and continuous on [a, b]. 2. Explain why it is enough to prove there exists c in (a, b) such that F (c) = Explain why F has a global maximum and a global minimum on [a, b]. 4. Explain why if either the global maximum or the global minimum occur at an interior point c of [a, b], then F (c) = Therefore, we must show that at least one of these (the global maximum or global minimum) occurs at an interior point. Prove it by contradiction. Assume they do not, that is they both happen at the end points. Show this leads to a contradiction. Note that there are two cases to consider. Case 1: F (a) is a global maximum, F (b) is a global minimum. Compute the sign of f (b) C using the fact that f (b) C = F (b) and compare what you get with the fact that f (a) < C < f (b). Case 2: F (a) is a global minimum, F (b) is a global maximum. Do the same as for case 1, except use f (a) C. Remark The above theorem can be used to determine where a function f is increasing or decreasing. From corollary 6.2.9, we know that it is enough to know the sign of f (x). If we know that there exist two numbers c 1 and c 2 with c 1 < c 2 such that f (c 1 ) = f (c 2 ) = 0 and we also know that f (x) 0

3 6.2. IMPORTANT THEOREMS 225 for every x (c 1, c 2 ) then this theorem tells us that f (c) is either positive or negative on the whole interval (c 1, c 2 ) thus to know the sign of f (x) on (c 1, c 2 ), it is enough to know it at a point of (c 2, c 2 ) Mean Value Theorems These theorems are among the most useful theorems not only for the applications they have, but also for their use in establishing other important results. We begin with the most general version of the mean value theorem. Then, we will derive the Mean Value Theorem usually taught in differential calculus from it. Theorem (Cauchy s Mean Value Theorem) Let f and g be two functions which are continuous on [a, b] and differentiable on (a, b). Then, there exists c in (a, b) such that or [f (b) f (a)] g (c) = [g (b) g (a)] f (c) f (b) f (a) g (b) g (a) = f (c) g (c) Proof. We outline the proof and ask the reader to fill in the details as part of the homework. Define F : [a, b] R by F (x) = [f (b) f (a)] g (x) [g (b) g (a)] f (x). Explain why F is continuous on [a, b] and differentiable on (a, b). Show that F (a) = F (b). Explain why to prove the result it is enough to show that there exists c in (a, b) such that F (c) = 0. There are two cases to consider: If F is a constant function, any c in (a, b) will work. If F is not constant, use Fermat s theorem to show that F (c) = 0 for some c. Be careful, you will have to establish that c is an interior point in order to apply Fermat s theorem. Theorem (Mean Value Theorem) If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f (c) = f (b) f (a) b a Proof. Apply Cauchy s mean value theorem with g (x) = x.

4 226 CHAPTER 6. DIFFERENTIATION The geometric interpretation of the Mean Value Theorem is that there exists a point c at which the slope of the tangent is the same as the slope of the secant line through (a, f (a)) and (b, f (b)). This can be seen easily on a picture. Cauchy s Mean Value Theorem has the same geometric interpretation when we consider the curves as given parametrically by x = f (t), and y = g (t). The Mean Value Theorem allows us to link the derivative of a function with the function itself. This has immediate consequences. Corollary Suppose that f is continuous on [a, b] and diff erentiable on (a, b). Then: 1. If f (x) = 0 for all x in (a, b), then f is a constant function on [a, b]. 2. If f (x) 0 for all x in (a, b), then f is monotone increasing on [a, b]. 3. If f (x) > 0 for all x in (a, b), then f is strictly increasing on [a, b]. 4. If f (x) 0 for all x in (a, b), then f is monotone decreasing on [a, b]. 5. If f (x) < 0 for all x in (a, b), then f is strictly decreasing on [a, b]. Proof. Left as an exercise. Remark It is important to understand that the results of the corollary are only valid on an interval, not at a point. It is possible for a function to satisfy f (c) > 0 yet for f not to be increasing in any neighborhood of c as shown in problem 11. Remark Another important consequence of the Mean Value Theorem appears in the exercises. If a function has a bounded derivative on an interval, then it is uniformly continuous there. This fact can be used to show that sin x and cos x are uniformly continuous. Another application appears below. Example If a < b, prove that e a (b a) < e b e a < e b (b a) Let f (x) = e x. Then, by the Mean Value Theorem there exists a number c such that f (b) f (a) = f (c) (b a). So, e b e a = e c (b a) If a < b, then e a < e b since e x is monotone increasing. Since b a > 0, we have e a (b a) < e c (b a) < e b (b a) or e a (b a) < e b e a < e b (b a)

5 6.2. IMPORTANT THEOREMS Exercises 1. Sketch the graph of a function which satisfies the conditions in each case below. (a) f has a local maximum at x = 2, f (2) exists. (b) f has a local minimum at x = 2, f (2) exists. (c) f has a local maximum at x = 2, f (2) does not exists. (d) f has a local minimum at x = 2, f (2) does not exists. (e) f has a critical number at x = 2 (f (2) = 0) but f has no extrema at x = 2. (f) f has a critical number at x = 2 (f (2) does not exist) but f has no extrema at x = Complete the proof of theorem in the case f has a local maximum at c then do it in the case f has a local minimum at c. 3. Finish proving theorem using the hint, then do thoroughly the case f (a) > f (b). 4. Prove theorem using the hint provided. 5. Prove theorem using the hint provided. 6. Prove the following theorem known as Rolle s theorem: If f is continuous on [a, b], differentiable on (a, b) and f (a) = f (b), then there exists c (a, b) such that f (c) = Find a proof for theorem different than the one suggested in the notes. 8. Prove corollary Prove that if f has a bounded derivative on an interval I, then f is uniformly continuous on I. 10. Let f : [0, 2] R be continuous on [0, 2] and differentiable on (0, 2) with f (0) = 0 and f (1) = f (2) = 1. (a) Show that there is a c 1 in (0, 1) such that f (c 1 ) = 1. (b) Show that there is a c 2 in (1, 2) such that f (c 2 ) = 0. (c) Show that there is a c 3 in (0, 2) such that f (c 3 ) = Let f (x) = { x + 2x 2 sin 1 x if x 0 0 if x = 0. (a) Show that f (0) = 1

6 228 CHAPTER 6. DIFFERENTIATION (b) Show that f (x) takes on both positive and negative values in every neighborhood of Using some of the theorems in this section, outline and justify a step by step method used to find the local extreme values of a function on an open interval and the global extreme values of a function on a closed interval. Illustrate your technique with examples. 13. The following result is known as the second derivative test. Let f : [a, b] R be differentiable on (a, b). Let c (a, b) such that f (c) = 0. (a) If f (c) > 0, prove that f has a local minimum at c. (b) If f (c) < 0, prove that f has a local maximum at c. (c) Show by example that if f (c) = 0 then no conclusion can be made. 14. Let f : [a, b] R such that f is continuous on [a, b] and differentiable on (a, b). Suppose further that f (x) M on (a, b) for some positive constant M. (a) Prove that for every s, t in [a, b], f (s) f (t) M s t, in other words f is Lipschitz on [a, b]. (b) Make a conjecture about the value of M and prove your conjecture. (c) Under the conditions of this problem, prove that f is uniformly continuous on [a, b].

7 6.2. IMPORTANT THEOREMS Hints for the Exercises 1. Sketch the graph of a function which satisfies the conditions in each case below. (a) f has a local maximum at x = 2, f (2) exists. (b) f has a local minimum at x = 2, f (2) exists. (c) f has a local maximum at x = 2, f (2) does not exists. (d) f has a local minimum at x = 2, f (2) does not exists. (e) f has a critical number at x = 2 (f (2) = 0) but f has no extrema at x = 2. (f) f has a critical number at x = 2 (f (2) does not exist) but f has no extrema at x = Complete the proof of theorem in the case f has a local maximum at c then do it in the case f has a local minimum at c. Hint: outline of the proof for a local maximum. Write the definition of f having a local maximum at an interior point c of I. f (x) f (c) Study the sign of on each side of c. x c Use the previous step to deduce the sign of the one-sided derivatives. Use the fact that f is differentiable at c, to finish the proof. 3. Finish proving theorem using the hint, then do thoroughly the case f (a) > f (b). Hint: see the question. 4. Prove theorem using the hint provided. Hint: see the question. 5. Prove theorem using the hint provided. Hint: see the question. 6. Prove the following theorem known as Rolle s theorem: If f is continuous on [a, b], differentiable on (a, b) and f (a) = f (b), then there exists c (a, b) such that f (c) = 0. Hint: use theorem Find a proof for theorem different than the one suggested in the notes Hint: use Rolle s theorem on the function g (x) = f (x) f (a) f(b) f(a) b a (x a). 8. Prove corollary Hint: use the mean value theorem.

8 230 CHAPTER 6. DIFFERENTIATION 9. Prove that if f has a bounded derivative on an interval I, then f is uniformly continuous on I. 10. Let f : [0, 2] R be continuous on [0, 2] and differentiable on (0, 2) with f (0) = 0 and f (1) = f (2) = 1. Hint: use the mean value theorem. (a) Show that there is a c 1 in (0, 1) such that f (c 1 ) = 1. (b) Show that there is a c 2 in (1, 2) such that f (c 2 ) = 0. (c) Show that there is a c 3 in (0, 2) such that f (c 3 ) = Let f (x) = { x + 2x 2 sin 1 x if x 0 0 if x = 0. (a) Show that f (0) = 1 (b) Show that f (x) takes on both positive and negative values in every neighborhood of 0. Hint: for an arbitrary δ > 0, find 2 sequences (u n ) and (v n ), which are eventually in ( δ, δ) such that f (u n ) < 0 and f (v n ) > Using some of the theorems in this section, outline and justify a step by step method used to find the local extreme values of a function on an open interval and the global extreme values of a function on a closed interval. Illustrate your technique with examples. 13. The following result is known as the second derivative test. Let f : [a, b] R be differentiable on (a, b). Let c (a, b) such that f (c) = 0. Hint: use definitions and study the sign of f (x) on each side of c. (a) If f (c) > 0, prove that f has a local minimum at c. (b) If f (c) < 0, prove that f has a local maximum at c. (c) Show by example that if f (c) = 0 then no conclusion can be made. 14. Let f : [a, b] R such that f is continuous on [a, b] and differentiable on (a, b). Suppose further that f (x) M on (a, b) for some positive constant M. (a) Prove that for every s, t in [a, b], f (s) f (t) M s t, in other words f is Lipschitz on [a, b]. Hint: use the Mean Value Theorem. (b) Make a conjecture about the value of M and prove your conjecture. Hint: What can be said about the set of values {f (x) : x [a, b]}? (c) Under the conditions of this problem, prove that f is uniformly continuous on [a, b]. Hint: Use results in the section on uniformly continuous functions.

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