Rolle's Theorem

Rolle's theorem can be found in any calculus book, see for example Larson [4, p.168]. For an on-line reference go to http://www.math.hmc.edu/calculus/tutorials/mean_value/. the theorem is usually stated as follows:

If f(x) is continuous for a ≤ x ≤ b, and f ′(x) exists for a < x < b, and f(a) = f(b), then f ′(c) = 0 for at least one value of c with a < c < b.

On the previous page, the a and b are roots of a polynomial p(x), and in particular, p(a) = p(b) = 0. These points correspond to red dots on the x-axis of the figures for Rolle's theorem, although in the general case, the dots could equally well lie on any horizontal line, not just the x-axis.

The proof of Rolle's theorem depends on the fact that under the stated hypotheses, at a point where f achieves a maximum or minimum value, the derivative will equal 0. This is illustrated in Figure 1 below, where the red horizontal tangent line indicates a point where the derivative is 0.

Figure 1. At maxima and minima, the derivative is 0.

In outline the proof goes like this. Either the function is constant between a and b or it isn't. If it is constant, the derivative is 0 at each point, so we can choose any point to be the c required by the theorem. On the other hand, if the function is not constant, it must either include points above or below the common value of y = f(a) = f(b). That implies that there will either be a maximum value above y, where the derivative must then equal 0, or a minimum value below y, where again the derivative must equal 0. Note that the existence of a maximum or minimum value is a consequence of another calculus result, the extreme value theorem: a continuous function on a closed bounded interval assumes both a maximum value and a minimum value. see [4, p.168]