Extreme Value Theorem

If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem.

The standard proof of the first proceeds by noting that f is the continuous image of a compact set on the interval [a,b], so it must itself be compact. Since [a,b] is compact, it follows that the image f([a,b]) must also be compact.

See also

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Portions of this entry contributed by John Renze

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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

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Renze, John and Weisstein, Eric W. "Extreme Value Theorem." From MathWorld--A Wolfram Web Resource.

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