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

THEOREM OF MEAN VALUE. INDETERMINATE FORMS

Continuous curve.

105. Rolle's Theorem. 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.