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Rolle's Theorem

Let f be differentiable on the open interval (a,b) and continuous on the closed interval [a,b]. Then if f(a)=f(b), then there is at least one point c in (a,b) where f^'(c)=0.

Note that in elementary texts, the additional (but superfluous) condition f(a)=f(b)=0 is sometimes added (e.g., Anton 1999, p. 260).

See also

Fixed Point Theorem, Mean-Value Theorem

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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.

Cite this as:

Weisstein, Eric W. "Rolle's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RollesTheorem.html

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