extreme value theorem

Extreme Value TheoremMathworldPlanetmath. Let a and b be real numbers with a<b, and let f be a continuousMathworldPlanetmathPlanetmath, real valued function on [a,b]. Then there exists c,d[a,b] such that f(c)f(x)f(d) for all x[a,b].

Proof. We show only the existence of d. By the boundedness theorem f([a,b]) is bounded above; let l be the least upper bound of f([a,b]). Suppose, for a contradictionMathworldPlanetmathPlanetmath, that there is no d[a,b] such that f(d)=l. Then the function


is well defined and continuous on [a,b]. Since l is the least upper bound of f([a,b]), for any positive real number M we can find α[a,b] such that f(α)>l-1M, then


So g is unboundedPlanetmathPlanetmath on [a,b]. But by the boundedness theorem g is boundedPlanetmathPlanetmathPlanetmath on [a,b]. This contradiction finishes the proof.

Title extreme value theorem
Canonical name ExtremeValueTheorem
Date of creation 2013-03-22 14:29:21
Last modified on 2013-03-22 14:29:21
Owner classicleft (5752)
Last modified by classicleft (5752)
Numerical id 7
Author classicleft (5752)
Entry type Theorem
Classification msc 26A06
Synonym Weierstrass extreme value theorem