4. Applications of Derivatives

# 4.4 The Mean Value Theorem

### Learning Objectives

- Explain the meaning of Rolle’s theorem.
- Describe the significance of the Mean Value Theorem.
- State three important consequences of the Mean Value Theorem.

The **Mean Value Theorem** is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem.

# Rolle’s Theorem

Informally, **Rolle’s theorem** states that if the outputs of a differentiable function are equal at the endpoints of an interval, then there must be an interior point where (Figure) illustrates this theorem.

### Rolle’s Theorem

Let be a continuous function over the closed interval and differentiable over the open interval such that There then exists at least one such that

## Proof

Let We consider three cases:

- for all
- There exists such that
- There exists such that

Case 1: If for all then for all

Case 2: Since is a continuous function over the closed, bounded interval by the extreme value theorem, it has an absolute maximum. Also, since there is a point such that the absolute maximum is greater than Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point Because has a maximum at an interior point and is differentiable at by Fermat’s theorem,

Case 3: The case when there exists a point such that is analogous to case 2, with maximum replaced by minimum.

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An important point about Rolle’s theorem is that the differentiability of the function is critical. If is not differentiable, even at a single point, the result may not hold. For example, the function is continuous over and but for any as shown in the following figure.

Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points where

### Using Rolle’s Theorem

For each of the following functions, verify that the function satisfies the criteria stated in Rolle’s theorem and find all values in the given interval where

- over
- over

#### Solution

- Since is a polynomial, it is continuous and differentiable everywhere. In addition, Therefore, satisfies the criteria of Rolle’s theorem. We conclude that there exists at least one value such that Since we see that implies as shown in the following graph.
- As in part a. is a polynomial and therefore is continuous and differentiable everywhere. Also, That said, satisfies the criteria of Rolle’s theorem. Differentiating, we find that Therefore, when Both points are in the interval and, therefore, both points satisfy the conclusion of Rolle’s theorem as shown in the following graph.

Verify that the function defined over the interval satisfies the conditions of Rolle’s theorem. Find all points guaranteed by Rolle’s theorem.

#### Solution

# The Mean Value Theorem and Its Meaning

Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions that are zero at the endpoints. The Mean Value Theorem generalizes Rolle’s theorem by considering functions that are not necessarily zero at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem ((Figure)). The Mean Value Theorem states that if is continuous over the closed interval and differentiable over the open interval then there exists a point such that the tangent line to the graph of at is parallel to the secant line connecting and

### Mean Value Theorem

Let be continuous over the closed interval and differentiable over the open interval Then, there exists at least one point such that

## Proof

The proof follows from Rolle’s theorem by introducing an appropriate function that satisfies the criteria of Rolle’s theorem. Consider the line connecting and Since the slope of that line is

and the line passes through the point the equation of that line can be written as

Let denote the vertical difference between the point and the point on that line. Therefore,

Since the graph of intersects the secant line when and we see that Since is a differentiable function over is also a differentiable function over Furthermore, since is continuous over is also continuous over Therefore, satisfies the criteria of Rolle’s theorem. Consequently, there exists a point such that Since

we see that

Since we conclude that

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In the next example, we show how the Mean Value Theorem can be applied to the function over the interval The method is the same for other functions, although sometimes with more interesting consequences.

### Verifying that the Mean Value Theorem Applies

For over the interval show that satisfies the hypothesis of the Mean Value Theorem, and therefore there exists at least one value such that is equal to the slope of the line connecting and Find these values guaranteed by the Mean Value Theorem.

#### Solution

We know that is continuous over and differentiable over Therefore, satisfies the hypotheses of the Mean Value Theorem, and there must exist at least one value such that is equal to the slope of the line connecting and ((Figure)). To determine which value(s) of are guaranteed, first calculate the derivative of The derivative The slope of the line connecting and is given by

We want to find such that That is, we want to find such that

Solving this equation for we obtain At this point, the slope of the tangent line equals the slope of the line joining the endpoints.

One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let and denote the position and velocity of the car, respectively, for h. Assuming that the position function is differentiable, we can apply the Mean Value Theorem to conclude that, at some time the speed of the car was exactly

### Mean Value Theorem and Velocity

If a rock is dropped from a height of 100 ft, its position seconds after it is dropped until it hits the ground is given by the function

- Determine how long it takes before the rock hits the ground.
- Find the average velocity of the rock for when the rock is released and the rock hits the ground.
- Find the time guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is

#### Solution

- When the rock hits the ground, its position is Solving the equation for we find that Since we are only considering the ball will hit the ground sec after it is dropped.
- The average velocity is given by
- The instantaneous velocity is given by the derivative of the position function. Therefore, we need to find a time such that Since is continuous over the interval and differentiable over the interval by the Mean Value Theorem, there is guaranteed to be a point such that
Taking the derivative of the position function we find that Therefore, the equation reduces to Solving this equation for we have Therefore, sec after the rock is dropped, the instantaneous velocity equals the average velocity of the rock during its free fall: -40 ft/sec.

Suppose a ball is dropped from a height of 200 ft. Its position at time is Find the time when the instantaneous velocity of the ball equals its average velocity.

#### Solution

sec

#### Hint

First, determine how long it takes for the ball to hit the ground. Then, find the average velocity of the ball from the time it is dropped until it hits the ground.

# Corollaries of the Mean Value Theorem

Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections.

At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if for all in some interval then is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly.

### Corollary 1: Functions with a Derivative of Zero

Let be differentiable over an interval If for all then constant for all

## Proof

Since is differentiable over must be continuous over Suppose is not constant for all in Then there exist where and Choose the notation so that Therefore,

Since is a differentiable function, by the Mean Value Theorem, there exists such that

Therefore, there exists such that which contradicts the assumption that for all

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From (Figure), it follows that if two functions have the same derivative, they differ by, at most, a constant.

### Corollary 2: Constant Difference Theorem

If and are differentiable over an interval and for all then for some constant

## Proof

Let Then, for all By Corollary 1, there is a constant such that for all Therefore, for all

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The third corollary of the Mean Value Theorem discusses when a function is increasing and when it is decreasing. Recall that a function is increasing over if whenever whereas is decreasing over if whenever Using the Mean Value Theorem, we can show that if the derivative of a function is positive, then the function is increasing; if the derivative is negative, then the function is decreasing ((Figure)). We make use of this fact in the next section, where we show how to use the derivative of a function to locate local maximum and minimum values of the function, and how to determine the shape of the graph.

This fact is important because it means that for a given function if there exists a function such that then, the only other functions that have a derivative equal to are for some constant We discuss this result in more detail later in the chapter.

### Corollary 3: Increasing and Decreasing Functions

Let be continuous over the closed interval and differentiable over the open interval

- If for all then is an increasing function over
- If for all then is a decreasing function over

## Proof

We will prove i.; the proof of ii. is similar. Suppose is not an increasing function on Then there exist and in such that but Since is a differentiable function over by the Mean Value Theorem there exists such that

Since we know that Also, tells us that We conclude that

However, for all This is a contradiction, and therefore must be an increasing function over

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### Key Concepts

- If is continuous over and differentiable over and then there exists a point such that This is Rolle’s theorem.
- If is continuous over and differentiable over then there exists a point such that
This is the Mean Value Theorem.

- If over an interval then is constant over
- If two differentiable functions and satisfy over then for some constant
- If over an interval then is increasing over If over then is decreasing over

**1.** Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.

**2.** Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

#### Solution

One example is

**3.** When are Rolle’s theorem and the Mean Value Theorem equivalent?

**4.** If you have a function with a discontinuity, is it still possible to have Draw such an example or prove why not.

#### Solution

Yes, but the Mean Value Theorem still does not apply

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

**5.**

**6.**

#### Solution

**7.**

**8.**

#### Solution

**9.**

For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points such that

**10. [T]** over

#### Solution

2 points

**11. [T]** over

**12. [T]** over

#### Solution

5 points

**13. [T]** over

For the following exercises, use the Mean Value Theorem and find all points such that

**14.**

#### Solution

**15.**

**16.**

#### Solution

**17.**

**18.**

#### Solution

**19.**

For the following exercises, show there is no such that Explain why the Mean Value Theorem does not apply over the interval

**20.**

#### Solution

Not differentiable

**21.**

**22.**

#### Solution

Not differentiable

**23.** (*Hint*: This is called the *floor function* and it is defined so that is the largest integer less than or equal to )

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval Justify your answer.

**24.** over

#### Solution

Yes

**25.** over

**26.** over

#### Solution

The Mean Value Theorem does not apply since the function is discontinuous at

**27.** over

**28.** over

#### Solution

Yes

**29.** over

**30.** over

#### Solution

The Mean Value Theorem does not apply; discontinuous at

**31.** over

**32.** over

#### Solution

Yes

**33.** over

**34.** over

#### Solution

The Mean Value Theorem does not apply; not differentiable at

For the following exercises, consider the roots of the equation.

**35.** Show that the equation has exactly one real root. What is it?

**36.** Find the conditions for exactly one root (double root) for the equation

#### Solution

**37.** Find the conditions for to have one root. Is it possible to have more than one root?

For the following exercises, use a calculator to graph the function over the interval and graph the secant line from to Use the calculator to estimate all values of as guaranteed by the Mean Value Theorem. Then, find the exact value of if possible, or write the final equation and use a calculator to estimate to four digits.

**38. [T]** over

#### Solution

[latex]c=\text{±}0.1533[/latex]

**39. [T]** over

**40. [T]** over

#### Solution

The Mean Value Theorem does not apply.

**41. [T]** over

**42. [T]** over

#### Solution

[latex]c=3.133,5.867[/latex]

**43.** At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?

**44.** Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.

#### Solution

Yes

**45.** Show that and have the same derivative. What can you say about

**46.** Show that and have the same derivative. What can you say about

#### Solution

It is constant.

## Glossary

- mean value theorem
- if is continuous over and differentiable over then there exists such that

- rolle’s theorem
- if is continuous over and differentiable over and if then there exists such that

## Hint

Find all values where