The Mean Value Theorem

Learning Outcomes

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

Rolle’s Theorem

Informally, Rolle’s theorem states that if the outputs of a differentiable function [latex]f[/latex] are equal at the endpoints of an interval, then there must be an interior point [latex]c[/latex] where [latex]f^{\prime}(c)=0[/latex]. Figure 1 illustrates this theorem.

The figure is divided into three parts labeled a, b, and c. Figure a shows the first quadrant with values a, c, and b marked on the x-axis. A downward-facing parabola is drawn such that its values at a and b are the same. The point c is the global maximum, and it is noted that f’(c) = 0. Figure b shows the first quadrant with values a, c, and b marked on the x-axis. An upward-facing parabola is drawn such that its values at a and b are the same. The point c is the global minimum, and it is noted that f’(c) = 0. Figure c shows the first quadrant with points a, c1, c2, and b marked on the x-axis. One period of a sine wave is drawn such that its values at a and b are equal. The point c1 is the global maximum, and it is noted that f’(c1) = 0. The point c2 is the global minimum, and it is noted that f’(c2) = 0.

Figure 1. If a differentiable function f satisfies [latex]f(a)=f(b)[/latex], then its derivative must be zero at some point(s) between [latex]a[/latex] and [latex]b[/latex].

Rolle’s Theorem

Let [latex]f[/latex] be a continuous function over the closed interval [latex][a,b][/latex] and differentiable over the open interval [latex](a,b)[/latex] such that [latex]f(a)=f(b)[/latex]. There then exists at least one [latex]c \in (a,b)[/latex] such that [latex]f^{\prime}(c)=0[/latex].

Proof

Let [latex]k=f(a)=f(b)[/latex]. We consider three cases:

  1. [latex]f(x)=k[/latex] for all [latex]x \in (a,b)[/latex].
  2. There exists [latex]x \in (a,b)[/latex] such that [latex]f(x)>k[/latex].
  3. There exists [latex]x \in (a,b)[/latex] such that [latex]f(x)<k[/latex].

Case 1: If [latex]f(x)=k[/latex] for all [latex]x \in (a,b)[/latex], then [latex]f^{\prime}(x)=0[/latex] for all [latex]x \in (a,b)[/latex].

Case 2: Since [latex]f[/latex] is a continuous function over the closed, bounded interval [latex][a,b][/latex], by the extreme value theorem, it has an absolute maximum. Also, since there is a point [latex]x \in (a,b)[/latex] such that [latex]f(x)>k[/latex], the absolute maximum is greater than [latex]k[/latex]. Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point [latex]c \in (a,b)[/latex]. Because [latex]f[/latex] has a maximum at an interior point [latex]c[/latex], and [latex]f[/latex] is differentiable at [latex]c[/latex], by Fermat’s theorem, [latex]f^{\prime}(c)=0[/latex].

Case 3: The case when there exists a point [latex]x \in (a,b)[/latex] such that [latex]f(x)<k[/latex] is analogous to case 2, with maximum replaced by minimum.

[latex]_\blacksquare[/latex]

An important point about Rolle’s theorem is that the differentiability of the function [latex]f[/latex] is critical. If [latex]f[/latex] is not differentiable, even at a single point, the result may not hold. For example, the function [latex]f(x)=|x|-1[/latex] is continuous over [latex][-1,1][/latex] and [latex]f(-1)=0=f(1)[/latex], but [latex]f^{\prime}(c) \ne 0[/latex] for any [latex]c \in (-1,1)[/latex] as shown in the following figure.

The function f(x) = |x| − 1 is graphed. It is shown that f(1) = f(−1), but it is noted that there is no c such that f’(c) = 0.

Figure 2. Since [latex]f(x)=|x|-1[/latex] is not differentiable at [latex]x=0[/latex], the conditions of Rolle’s theorem are not satisfied. In fact, the conclusion does not hold here; there is no [latex]c \in (-1,1)[/latex] such that [latex]f^{\prime}(c)=0[/latex].

Let’s now consider functions that satisfy the conditions of Rolle’s theorem and calculate explicitly the points [latex]c[/latex] where [latex]f^{\prime}(c)=0[/latex].

Example: 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 [latex]c[/latex] in the given interval where [latex]f^{\prime}(c)=0[/latex].

  1. [latex]f(x)=x^2+2x[/latex] over [latex][-2,0][/latex]
  2. [latex]f(x)=x^3-4x[/latex] over [latex][-2,2][/latex]

Try It

Verify that the function [latex]f(x)=2x^2-8x+6[/latex] defined over the interval [latex][1,3][/latex] satisfies the conditions of Rolle’s theorem. Find all points [latex]c[/latex] guaranteed by Rolle’s theorem.

Watch the following video to see the worked solution to Example: Using Rolle’s Theorem and the above Try It.

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 [latex]f[/latex]defined on a closed interval [latex][a,b][/latex] with [latex]f(a)=f(b)[/latex]. The Mean Value Theorem generalizes Rolle’s theorem by considering functions that do not necessarily have equal value at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem (Figure 5). The Mean Value Theorem states that if [latex]f[/latex] is continuous over the closed interval [latex][a,b][/latex] and differentiable over the open interval [latex](a,b)[/latex], then there exists a point [latex]c \in (a,b)[/latex] such that the tangent line to the graph of [latex]f[/latex] at [latex]c[/latex] is parallel to the secant line connecting [latex](a,f(a))[/latex] and [latex](b,f(b))[/latex].

A vaguely sinusoidal function y = f(x) is drawn. On the x-axis, a, c1, c2, and b are marked. On the y-axis, f(a) and f(b) are marked. The function f(x) starts at (a, f(a)), decreases to c1, increases to c2, and then decreases to (b, f(b)). A secant line is drawn between (a, f(a)) and (b, f(b)), and it is noted that this line has slope (f(b) – f(a))/(b − a). The tangent lines at c1 and c2 are drawn, and these lines are parallel to the secant line. It is noted that the slopes of these tangent lines are f’(c1) and f’(c2), respectively.

Figure 5. The Mean Value Theorem says that for a function that meets its conditions, at some point the tangent line has the same slope as the secant line between the ends. For this function, there are two values [latex]c_1[/latex] and [latex]c_2[/latex] such that the tangent line to [latex]f[/latex] at [latex]c_1[/latex] and [latex]c_2[/latex] has the same slope as the secant line.

Mean Value Theorem

Let [latex]f[/latex] be continuous over the closed interval [latex][a,b][/latex] and differentiable over the open interval [latex](a,b)[/latex]. Then, there exists at least one point [latex]c \in (a,b)[/latex] such that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

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 [latex](a,f(a))[/latex] and [latex](b,f(b))[/latex]. Since the slope of that line is

[latex]\dfrac{f(b)-f(a)}{b-a}[/latex]

 

and the line passes through the point [latex](a,f(a))[/latex], the equation of that line can be written as

[latex]y=\dfrac{f(b)-f(a)}{b-a}(x-a)+f(a)[/latex]

 

Let [latex]g(x)[/latex] denote the vertical difference between the point [latex](x,f(x))[/latex] and the point [latex](x,y)[/latex] on that line. Therefore,

[latex]g(x)=f(x)-\left[\dfrac{f(b)-f(a)}{b-a}(x-a)+f(a)\right][/latex]

 

A vaguely sinusoidal function y = f(x) is drawn. On the x-axis, a and b are marked. On the y-axis, f(a) and f(b) are marked. The function f(x) starts at (a, f(a)), decreases, then increases, and then decreases to (b, f(b)). A secant line is drawn between (a, f(a)) and (b, f(b)), and it is noted that this line has equation y = ((f(b) – f(a))/(b − a)) (x − a) + f(x). A line is drawn between the maximum of f(x) and the secant line and it is marked g(x).

Figure 6. The value [latex]g(x)[/latex] is the vertical difference between the point [latex](x,f(x))[/latex] and the point [latex](x,y)[/latex] on the secant line connecting [latex](a,f(a))[/latex] and [latex](b,f(b)).[/latex]

Since the graph of [latex]f[/latex] intersects the secant line when [latex]x=a[/latex] and [latex]x=b[/latex], we see that [latex]g(a)=0=g(b)[/latex]. Since [latex]f[/latex] is a differentiable function over [latex](a,b)[/latex], [latex]g[/latex] is also a differentiable function over [latex](a,b)[/latex]. Furthermore, since [latex]f[/latex] is continuous over [latex][a,b][/latex], [latex]g[/latex] is also continuous over [latex][a,b][/latex]. Therefore, [latex]g[/latex] satisfies the criteria of Rolle’s theorem. Consequently, there exists a point [latex]c \in (a,b)[/latex] such that [latex]g^{\prime}(c)=0[/latex]. Since

[latex]g^{\prime}(x)=f^{\prime}(x)-\dfrac{f(b)-f(a)}{b-a}[/latex],

 

we see that

[latex]g^{\prime}(c)=f^{\prime}(c)-\dfrac{f(b)-f(a)}{b-a}[/latex]

 

Since [latex]g^{\prime}(c)=0[/latex], we conclude that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

[latex]_\blacksquare[/latex]

In the next example, we show how the Mean Value Theorem can be applied to the function [latex]f(x)=\sqrt{x}[/latex] over the interval [latex][0,9][/latex]. The method is the same for other functions, although sometimes with more interesting consequences.

Example: Verifying that the Mean Value Theorem Applies

For [latex]f(x)=\sqrt{x}[/latex] over the interval [latex][0,9][/latex], show that [latex]f[/latex] satisfies the hypothesis of the Mean Value Theorem, and therefore there exists at least one value [latex]c \in (0,9)[/latex] such that [latex]f^{\prime}(c)[/latex] is equal to the slope of the line connecting [latex](0,f(0))[/latex] and [latex](9,f(9))[/latex]. Find these values [latex]c[/latex] guaranteed by the Mean Value Theorem.

One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 hr down a straight road with an average velocity of 45 mph. Let [latex]s(t)[/latex] and [latex]v(t)[/latex] denote the position and velocity of the car, respectively, for [latex]0 \le t \le 1[/latex] hr. Assuming that the position function [latex]s(t)[/latex] is differentiable, we can apply the Mean Value Theorem to conclude that, at some time [latex]c \in (0,1)[/latex], the speed of the car was exactly

[latex]v(c)=s^{\prime}(c)=\dfrac{s(1)-s(0)}{1-0}=45[/latex] mph

 

Example: Mean Value Theorem and Velocity

If a rock is dropped from a height of 100 ft, its position [latex]t[/latex] seconds after it is dropped until it hits the ground is given by the function [latex]s(t)=-16t^2+100[/latex].

  1. Determine how long it takes before the rock hits the ground.
  2. Find the average velocity [latex]v_{\text{avg}}[/latex] of the rock for when the rock is released and the rock hits the ground.
  3. Find the time [latex]t[/latex] guaranteed by the Mean Value Theorem when the instantaneous velocity of the rock is [latex]v_{\text{avg}}[/latex].

Watch the following video to see the worked solution to Example: Mean Value Theorem and Velocity.

Try It

Suppose a ball is dropped from a height of 200 ft. Its position at time [latex]t[/latex] is [latex]s(t)=-16t^2+200[/latex]. Find the time [latex]t[/latex] when the instantaneous velocity of the ball equals its average velocity.

Try It

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 [latex]f^{\prime}(x)=0[/latex] for all [latex]x[/latex] in some interval [latex]I[/latex], then [latex]f(x)[/latex] 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 [latex]f[/latex] be differentiable over an interval [latex]I[/latex]. If [latex]f^{\prime}(x)=0[/latex] for all [latex]x \in I[/latex], then [latex]f(x)[/latex] is constant for all [latex]x \in I[/latex].

Proof

Since [latex]f[/latex] is differentiable over [latex]I[/latex], [latex]f[/latex] must be continuous over [latex]I[/latex]. Suppose [latex]f(x)[/latex] is not constant for all [latex]x[/latex] in [latex]I[/latex]. Then there exist [latex]a,b \in I[/latex], where [latex]a \ne b[/latex] and [latex]f(a) \ne f(b)[/latex]. Choose the notation so that [latex]a<b[/latex]. Therefore,

[latex]\dfrac{f(b)-f(a)}{b-a} \ne 0[/latex]

 

Since [latex]f[/latex] is a differentiable function, by the Mean Value Theorem, there exists [latex]c \in (a,b)[/latex] such that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

 

Therefore, there exists [latex]c \in I[/latex] such that [latex]f^{\prime}(c) \ne 0[/latex], which contradicts the assumption that [latex]f^{\prime}(x)=0[/latex] for all [latex]x \in I[/latex].

[latex]_\blacksquare[/latex]

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

Corollary 2: Constant Difference Theorem

If [latex]f[/latex] and [latex]g[/latex] are differentiable over an interval [latex]I[/latex] and [latex]f^{\prime}(x)=g^{\prime}(x)[/latex] for all [latex]x \in I[/latex], then [latex]f(x)=g(x)+C[/latex] for some constant [latex]C[/latex].

Proof

Let [latex]h(x)=f(x)-g(x)[/latex]. Then, [latex]h^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x)=0[/latex] for all [latex]x \in I[/latex]. By Corollary 1, there is a constant [latex]C[/latex] such that [latex]h(x)=C[/latex] for all [latex]x \in I[/latex]. Therefore, [latex]f(x)=g(x)+C[/latex] for all [latex]x \in I[/latex].

[latex]_\blacksquare[/latex]

The third corollary of the Mean Value Theorem discusses when a function is increasing and when it is decreasing. Recall that a function [latex]f[/latex] is increasing over [latex]I[/latex] if [latex]f(x_1)<f(x_2)[/latex] whenever [latex]x_1<x_2[/latex], whereas [latex]f[/latex] is decreasing over [latex]I[/latex] if [latex]f(x_1)>f(x_2)[/latex] whenever [latex]x_1<x_2[/latex]. 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 9). 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 [latex]f[/latex], if there exists a function [latex]F[/latex] such that [latex]F^{\prime}(x)=f(x)[/latex]; then, the only other functions that have a derivative equal to [latex]f[/latex] are [latex]F(x)+C[/latex] for some constant [latex]C[/latex]. We discuss this result in more detail later in the chapter.

A vaguely sinusoidal function f(x) is graphed. It increases from somewhere in the second quadrant to (a, f(a)). In this section it is noted that f’ > 0. Then in decreases from (a, f(a)) to (b, f(b)). In this section it is noted that f’ < 0. Finally, it increases to the right of (b, f(b)) and it is noted in this section that f’ > 0

Figure 9. If a function has a positive derivative over some interval [latex]I[/latex], then the function increases over that interval [latex]I[/latex]; if the derivative is negative over some interval [latex]I[/latex], then the function decreases over that interval [latex]I[/latex].

Corollary 3: Increasing and Decreasing Functions

Let [latex]f[/latex] be continuous over the closed interval [latex][a,b][/latex] and differentiable over the open interval [latex](a,b)[/latex].

  1. If [latex]f^{\prime}(x)>0[/latex] for all [latex]x \in (a,b)[/latex], then [latex]f[/latex] is an increasing function over [latex][a,b][/latex].
  2. If [latex]f^{\prime}(x)<0[/latex] for all [latex]x \in (a,b)[/latex], then [latex]f[/latex] is a decreasing function over [latex][a,b][/latex].

Proof

We will prove 1.; the proof of 2. is similar. Suppose [latex]f[/latex] is not an increasing function on [latex]I[/latex]. Then there exist [latex]a[/latex] and [latex]b[/latex] in [latex]I[/latex] such that [latex]a<b[/latex], but [latex]f(a) \ge f(b)[/latex]. Since [latex]f[/latex] is a differentiable function over [latex]I[/latex], by the Mean Value Theorem there exists [latex]c \in (a,b)[/latex] such that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex]

 

Since [latex]f(a) \ge f(b)[/latex], we know that [latex]f(b)-f(a) \le 0[/latex]. Also, [latex]a<b[/latex] tells us that [latex]b-a>0[/latex]. We conclude that

[latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a} \le 0[/latex]

 

However, [latex]f^{\prime}(x)>0[/latex] for all [latex]x \in I[/latex]. This is a contradiction, and therefore [latex]f[/latex] must be an increasing function over [latex]I[/latex].

[latex]_\blacksquare[/latex]