proof of least and greatest value of function

f is continuousMathworldPlanetmath, so it will transform compact sets into compact sets. Thus since [a,b] is compact, f([a,b]) is also compact. f will thus attain on the interval [a,b] a maximum and a minimum value because real compact sets are closed and boundedPlanetmathPlanetmathPlanetmathPlanetmath.

Consider the maximum and later use the same argument for -f to consider the minimum.

By a known theorem ( if the maximum is attained in the interior of the domain, c]a,b[ then f(c)is a maximumf(c)=0, since f is differentiableMathworldPlanetmathPlanetmath.

If the maximum isn’t attained in ]a,b[ and since it must be attained in [a,b] either f(a) or f(b) is a maximum.

For the minimum consider -f and note that -f will verify all conditions of the theorem and that a maximum of -f corresponds to a minimum of f and that -f(c)=0f(c)=0.

Title proof of least and greatest value of function
Canonical name ProofOfLeastAndGreatestValueOfFunction
Date of creation 2013-03-22 15:52:09
Last modified on 2013-03-22 15:52:09
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 5
Author cvalente (11260)
Entry type Proof
Classification msc 26B12
Related topic FermatsTheoremStationaryPoints
Related topic HeineBorelTheorem
Related topic CompactnessIsPreservedUnderAContinuousMap