If a real-valued function $f$ is continuous on some closed interval $[a,b]$, has at each interior point of this interval a finite derivative or an infinite derivative of definite sign and at the end points of this interval takes equal values, then in the open interval $(a,b)$ there exists at least one point at which the derivative of $f$ vanishes.
The geometric sense of Rolle's theorem is that on the graph of a function $f$ satisfying the requirements of the theorem there exists a point $(\xi,f(\xi))$, $a<\xi<b$, such that the tangent to the graph at this point is parallel to the $x$-axis.
The mechanical interpretation of Rolle's theorem is that for any material point moving continuously along a straight line and which has returned after a certain period of time to the initial point there exists an instant at which the instantaneous velocity has been zero.
This theorem was first obtained by M. Rolle  for algebraic polynomials.
|||M. Rolle, "Traité d'algèbre" , Paris (1690)|
|||S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)|
See also Finite-increments formula.
|[a1]||K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff|
|[a2]||T.M. Apostol, "Calculus" , 1 , Blaisdell (1967)|
Rolle theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rolle_theorem&oldid=32023