Theorem – Rolle’s theorem – Math-reference.net

Contents
1. Theorem
2. Proof
3. Consequences of the theorem.

Theorem

Let $f$ be a continuous function on $[a,b]$ that is differentiable on $(a,b)$ such that $f(a) = f(b) = 0$ . Then, there exists at least one point $c\in (a,b)$ such that $f'(c)=0$ .

Proof

For all $x \in [a,b]$ , there exists $x_1,x_2\in [a,b]$ such that $f(x_1) \leq f(x) \leq f(x_2)$ . If $x_1 = a$ and $x_2 = b$ , then by assumption $f(x) = 0$ , and hence $f'(0) = 0$ . Otherwise, $f$ has some maximum at a point $c\in (a,b)$ , and thus $f'(c) = 0$ . In either case, we have that such a point exists. $\square$