least and greatest value of function


If the real function f is

  1. 1.

    continuousMathworldPlanetmathPlanetmath on the closed intervalDlmfMathworldPlanetmath[a,b]  and

  2. 2.

    differentiableMathworldPlanetmathPlanetmath on the open interval(a,b),

then the functionMathworldPlanetmath has on the interval  [a,b]  a least value and a greatest value.  These are always got in the end of the interval or in the zero of the derivativePlanetmathPlanetmath.

Remark 1.  If the preconditions of the theorem are fulfilled by a function f, then one needs only to determine the values of f in the end points a and b of the interval and in the zeros of the derivative f inside the interval; then the least and the greatest value are found among those values.

Remark 2.  Note that the theorem does not require anything of the derivative f in the points a and b; one needs not even the right-sided derivative in a or the left-sided derivative in b.  Thus e.g. the function  f:x1-x2,  fulfilling the conditions of the theorem on the interval  [-1, 1]  but not having such one-sided derivatives, gains its least value in the end-point  x=-1  and its greatest value in the zero  x=0  of the derivative.

Remark 3.  The least value of a function is also called the absolute minimum and the greatest value the absolute maximum of the function.

Title least and greatest value of function
Canonical name LeastAndGreatestValueOfFunction
Date of creation 2013-03-22 15:38:57
Last modified on 2013-03-22 15:38:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 26B12
Synonym global extrema of real function
Related topic ExtremumMathworldPlanetmath
Related topic LeastAndGreatestNumber
Related topic FermatsTheoremStationaryPoints
Related topic MinimalAndMaximalNumber
Defines absolute minimum
Defines absolute maximum