Rolle theorem

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 [1] for algebraic polynomials.

References

[1]	M. Rolle, "Traité d'algèbre" , Paris (1690)
[2]	S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)

Comments

See also Finite-increments formula.

References

[a1]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff
[a2]	T.M. Apostol, "Calculus" , 1 , Blaisdell (1967)