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 $x$-axis (as at P); that is, the slope is zero.

Figure 13.1: Geometrically illustrating Rolle's theorem.

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 (see §8.3).

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

Figure 13.2: Counterexamples to Rolle's theorem.

Figure 13.2 (a) shows the graph of a function which is discontinuous (= $\infty$) for $x = c$, a value lying between $a$ and $b$. Figure 13.2 (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 the $x$-axis.