TheoremProved

Rolle's theorem

Statement

If $f:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $f(a)=f(b)$ , then there exists $c\in(a,b)$ with $f'(c)=0$ .

Why is it true?

A smooth path that returns to the exact height it started from must, at some point in between, have a moment of zero slope — the instant it stops rising and starts falling, or vice versa, at a peak or a valley.

Proof sketch

By the extreme value theorem, $f$ attains a maximum and a minimum on $[a,b]$ . If both occur only at the endpoints, then since $f(a)=f(b)$ , $f$ is constant on $[a,b]$ and $f'(c)=0$ for every $c\in(a,b)$ . Otherwise some extremum occurs at an interior point $c\in(a,b)$ ; since $f$ is differentiable there, the one-sided difference quotients approaching from both directions must agree and both be $\ge 0$ and $\le 0$ respectively (or vice versa for a minimum), forcing $f'(c)=0$ .

Topics that use this theorem

Applications of the Derivative

Related theorems

Mean value theorem (Lagrange)Extreme value theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Carl B. Boyer, Uta C. Merzbach (2011). A History of Mathematics
James Stewart (2015). Calculus