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Rolle's Theorem

Rolle's theorem guarantees a root of the derivative between two points where a differentiable real-valued function has equal values. Let f be a real-valued function that is 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).

For a real-rooted polynomial p of polynomial degree n>=1 with real coefficients, Rolle's theorem implies that the derivative p^' is also a real-rooted polynomial. If p has k distinct roots with multiplicities m_1, ..., m_k, then p^' retains m_i-1 copies of the ith root and has at least one root between successive distinct roots. This accounts for all n-1 roots of p^', with multiplicity included. When n=1, p^' is a nonzero constant, which is real-rooted as well.

See also

Fixed Point Theorem, Mean-Value Theorem, Real-Rooted Polynomial

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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.

Referenced on Wolfram|Alpha

Rolle's Theorem

Cite this as:

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

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