Question

What does Rolle's Theorem state?

Answer 1

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Rolle's Theorem was named after the French mathematician Michel Rolle (1652–1719), who proposed the first known formal proof of it in 1691. The name "Rolle's theorem" was first appeared in 1834.

In calculus, Rolle's theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them — that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero.

Rolle's Theorem gives conditions that guarantee the existence of an extreme value in the interior of a closed interval.

The Extreme Value Theorem states that a continuous function on a closed interval must have both a minimum and a maximum on the interval. Let's consider the case when both of these values can occur at the endpoints.

Rolle's Theorem

Let's consider a function $f$ which is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. If $f(a) = f(b)$ then there is at least one number $c$ in $(a, b)$ such that $f'(c) = 0$.

Proof

Let $f(a) = q = f(b)$.

Case 1. If $f(x) = q$ for $\forall x$ in $[a, b] \Rightarrow f(x)$ is constant on the interval, then by the Constant Rule Theorem, which states that the derivative of a constant function is 0, $f'(x) = 0$ for $\forall x$ in $(a, b)$.

Case 2. Assume that $f(x) > q$ for some value of $x$ in $(a, b)$. Then by the Extreme Value Theorem $f(x)$ has a maximum at some $c$ in the interval. Moreover, because $f(x) > q$, this maximum does not occur at either endpoints. Therefore, $f(x)$ has a maximum in the open interval $(a, b)$. This implies that $f(c)$ is a relative maximum and, we can apply the Theorem about Relative Extrema which states that Relative Extrema occur only at critical numbers of $f(x)$. So, $c$ is a critical number of $f(x)$. And because $f(x)$ is differentiable function at $c \Rightarrow f'(c) = 0$.

Case 3. Assume that $f(x) < q$ for some value of $x$ in $(a, b)$. Then by the Extreme Value Theorem $f(x)$ has a minimum at some $c$ in the interval. Moreover, because $f(x) < q$, this minimum does not occur at either endpoints. Therefore, $f(x)$ has a minimum in the open interval $(a, b)$. This implies that $f(c)$ is a relative minimum and, we can apply the Theorem about Relative Extrema which states that Relative Extrema occur only at critical numbers of $f(x)$. So, $c$ is a critical number of $f(x)$. And because $f(x)$ is differentiable function at $c \Rightarrow f'(c) = 0$.

Geometric interpretation of Rolle's Theorem:

From Rolle's Theorem $\Rightarrow$ if a function is continuous on $[a, b]$ and differentiable on $(a, b)$ and if $f(a) = f(b)$ then there must be at least one value of $x$ between $a$ and $b$ at which the graph of has a horizontal tangent.

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