Rolle's Theorem

Let  be differentiable on the open interval  and continuous on the closed interval . Then if , then there is at least one point  where .

Note that in elementary texts, the additional (but superfluous) condition  is sometimes added (e.g., Anton 1999, p. 260).

REFERENCES:

Anton, H. "Rolle's Theorem; Mean Value Theorem." §4.9 in Calculus: A New Horizon, 6th ed. New York: Wiley, pp. 260-266, 1999.

Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, p. 184, 1967.