The Mean Value Theorem

The mean value theorem states:

If f is differentiable on the interval (a,b) and (at least) continuous at the endpoints, a and b, then

f(b) - f(a) = f '(c) (b - a)

for some number c between a and b.

The numerical value of c is of no interest whatsoever! All anyone cares about is that it exists.

The truth of the mean value theorem is fairly obvious from the traditional picture [Fig. 3, p. 191 of Stewart]. (Note the slopes of the secant and tangent lines.) The standard proof (see the textbook) reduces the problem to an important special case, Rolle's theorem (where the tangent line is horizontal); Rolle's theorem is proved from the extreme value theorem (a continuous function on a closed, finite interval takes on a maximum value and a minimum value), which is worth knowing for other purposes; finally, the extreme value theorem is usually declared to be too difficult to prove in an elementary course.

The main use of the mean value theorem is in proving other theorems.

• Most importantly, now we can settle some unfinished business in the basic theory of integration. To justify antiderivative formulas such as

  [integral] x² dx= (1/3) x³ + C

  and to prove "the fundamental theorem of calculus, part 2" we needed to assume this lemma:

  If f₁'(x) = f₂'(x) for all x in an interval, then f₁(x) = f₂(x) + C for all x in that interval.

  This is now proved on pp. 193-194 of Stewart by means of the mean value theorem.

• On p. 196, the mean value theorem is used to prove the basic fact that if the derivative of a function is positive, then the function is increasing. (This turns out to be less obvious than it seems at first sight.)

If you are not impressed by the mean value theorem, you are actually in good company. Whether traditional calculus books put too much emphasis on the mean value theorem is a frequent topic of debate among mathematics teachers. See, for instance, the debate on pp. 231-245 of the American Mathematical Monthly, Vol. 104, No. 3 (March, 1997).

In this course we have not gone in detail into the proofs of the mean value theorem and its many corollaries, following the dictum of one of America's foremost mathematicians and teachers, R. P. Boas:

"Only professional mathematicians learn anything from proofs. Other people learn from explanations."

It is important, nevertheless, for students to become accustomed to reading precise statements of theorems, and to avoid being overwhelmed or distracted by the technical conditions stated in them. Let's look again at the mean value theorem:

If f is

1. differentiable on the interval (a,b), and
2. continuous at the endpoints, a and b,

then

f(b) - f(a) = f '(c) (b - a)

for some number c between a and b.

Why do we need all the technical verbiage at the beginning -- the conditions (1) and (2)? Your reaction to such a situation should be to try to see how the statement could be false for a function that doesn't satisfy all the conditions.

Exercise

1. a function that is not continuous at an endpoint
2. a function that is not differentiable in the interior of the interval