The Mean Value Theorem

Before we discuss the Rolle’s theorem and the mean value theorem, we will begin by the following theorems on which our subject is based. Let’s begin by the following definition.

Definition

Consider a function f that is defined on a set S (which can be an interval) containing a point c. We say that:

a)     f(c) is the maximum of f on S if  for all .

b)     f(c) is the minimum of f on S if  for all .

c)     f(c) is an extreme value of f on S if it is either a maximum or a minimum of f on S.

It is not necessary that all functions have a maximum or a minimum value on a set. Consider the function  defined on . This function has neither a maximum nor a minimum value on the given interval. However, the following theorem is very useful.

The Maximum-minimum Theorem

Let the function f be continuous on a closed interval I. Then there are numbers a and b in that interval such that f(a) is the maximum value of f and f(b) is the minimum value of f on I.

The previous theorem guarantees the existence of at least one maximum and one minimum points, but there may be several maximum or minimum points. These points may be interior or exterior points of the interval. If they are interior points, the following theorem, which is due to Fermat, holds.

Fermat’s Theorem

If f(a) is an extreme value of f on an interval I, in which a is an interior point (we say that it is a local or a relative extreme value), then one of the following two statements is true:

a) .

b)  fails to exist.

NB

The converse of this theorem is not true. Consider the case of . The function does not have an extreme value at x=0. However, we have . There is the similar case of . Here,  does not exist, although the function does not have an extreme value at that point also. Thus, the theorem gives a necessary condition for a continuous function to have an extreme value. In the same time, this condition is not sufficient to prove that a function has an extreme value at a point.

Now, we will present Rolle’s theorem, which is a special case of the mean value theorem. The theorem, is given that name in honor of its discoverer, Michel Rolle, the French mathematician.

Rolle’s Theorem

Let f be a function continuous on  and differentiable on . Also, let . Then there is at least one number c, a < c < b, such that .

It is stressed that if the function is not differentiable at some point inside the given open interval, the theorem may not hold.

Now we can explain the mean value theorem. Geometrically, the mean value theorem says that if the graph of a continuous function has a tangent (which is not vertical) between two points A and B, then there is at least one point at which the tangent is parallel to the line . That can be written as the following.

The Mean Value Theorem

If f is continuous on  and differentiable on , there exists a number c, , such that:

.

Alternatively, .