Differentiation - Important Theorems

Transcription

1 Differentiation - Important Theorems Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

2 Introduction We study several important theorems related to differentiation These theorems are important from a theoretical point of view as well as for applications, in particular for finding extreme values It is assumed that at this point, the reader has reached enough maturity to be able to prove these theorems with a little guidance Guidance in the form of an outline for the proof of each theorem is what is provided The details are left to the reader and assigned as homework Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

3 Local Extrema and Fermat s Theorem Definition 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 Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

4 Local Extrema and Fermat s Theorem 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 Remember that not every critical number corresponds to an extreme value To find 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 Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

5 Intermediate Value Property of the Derivative Being differentiable implies being continuous, the converse is false However, the derivative itself needs not be continuous But 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 The above theorem can be used to find 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 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 ) Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

6 Mean Value Theorems Cauchy Mean Value Theorem 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 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 [f (b) f (a)] g (c) = [g (b) g (a)] f (c) or f (b) f (a) g (b) g (a) = f (c) g (c) Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

7 Mean Value Theorems The Mean Value Theorem 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 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) Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

8 Consequences of the Mean Value Theorems 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 differentiable 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] Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

9 Remarks About the Mean Value Theorems 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 the homework Another important consequence of the Mean Value Theorem also appears in the homework 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 Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10

10 Exercises See the problems at the end of section 72 in my notes on differentiation Philippe B Laval (KSU) Differentiation - Important Theorems Spring / 10