The extreme value theorem states that if a real-valued function is continuous on a bounded closed interval , then attains both a maximum and a minimum on . It is also called the Weierstrass extreme value theorem. The image of the compact set under the continuous function is a compact set in , so it contains its supremum and infimum.

The closed interval method uses this existence result to locate the global extrema.

See also

Closed Interval Method, Extremum Explore this topic in the MathWorld classroom

Portions of this entry contributed by John Renze

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References

Anton, H. Calculus with Analytic Geometry, 2nd ed. New York: Wiley, p. 229, 1984.Apostol, T. M. "The Extreme-Value Theorem for Continuous Functions." §3.16 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 150-152, 1967.Stewart, J. Single Variable Calculus, 6th ed. Belmont, CA: Brooks/Cole, p. 104, 2008.Stewart, J. Calculus: Early Transcendentals, 7th ed. Belmont, CA: Brooks/Cole, p. 275, 2012.

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Extreme Value Theorem

Cite this as:

Weisstein, Eric W., with contributions by John Renze. "Extreme Value Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExtremeValueTheorem.html

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