Rolle's theorem

Named after French mathematician Michel Rolle (1652–1719), although his 1691 proof covered only the case of polynomial functions and did not use the methods of differential calculus.

Rolle's theorem

1. (calculus) The theorem that any real-valued differentiable function that attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero. In mathematical terms, if ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ is differentiable on ${\displaystyle (a,b)}$ and ${\displaystyle f(a)=f(b)}$ then ${\displaystyle \exists c\in (a,b):f'(c)=0}$.