Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/188

 THEOREM OF MEAN VALUE. INDETERMINATE FORMS

Let $$y = f(x)$$ be a continuous single-valued function of x, vanishing for x = a and x = b, and suppose that $$f'(x)$$ changes continuously when x varies from a to b. The function will then be represented graphically by a continuous curve as in the figure. Geometric intuition shows us at once that for at least one value of x between a and b the tangent is parallel to the axis of X (as at P); that is, the slope is zero. This illustrates Rolle's Theorem:

If f(x) vanishes when x = a and x = b, and f(x) and f'(x) are continuous for all values of x from x = a to x = b, then f'(x) will be zero for at least one value of x between a and b.

This theorem is obviously true, because as x increases from a to b, $$f(x)$$ cannot always increase or always decrease as x increases, since $$f(a) = 0$$ and $$f(b) = 0$$. Hence for at least one value of x between a and b, $$f(x)$$ must cease to increase and begin to decrease, or else cease to decrease and begin to increase; and for that particular value of x the first derivative must be zero (§ 81).

That Rolle's Theorem does not apply when $$f(x)$$ or $$f'(x)$$ are discontinuous is illustrated as follows:

Fig. a shows the graph of a function which is discontinuous ($$= \infty$$) for $$x = c$$, a value lying between a and b. Fig. b shows a continuous function whose first derivative is discontinuous ($$= \infty$$) for such an intermediate value x = c. In either case it is seen that at no point on the graph between x = a and x = b does the tangent (or curve) be,come parallel to OX.