MATHEMATICAL ANALYSIS 07

Rolle's Theorem

If a smooth curve starts and ends at the same height, somewhere between those endpoints its tangent must be horizontal.

The theorem

Suppose \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and

\[f(a)=f(b).\]

Then there exists at least one \(c\in(a,b)\) such that

\[\boxed{f'(c)=0}\]

Why it is true

By the Extreme Value Theorem, the continuous function attains its extrema. If it is not constant, at least one relevant extremum occurs inside the interval; differentiability then gives derivative zero there.

Role in analysis

Rolle's theorem is a special case of the Mean Value Theorem and provides the key idea behind its proof.