least and greatest value of function

Theorem.

If the real function $f$ is

1. continuous on the closed interval  $[a,b]$  and

2. differentiable on the open interval  $(a,b)$,

then the function 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 derivative.

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 $f^{\prime }$ 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 $f^{\prime }$ 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  $f:x\mapsto \sqrt{1-x^2}$,  fulfilling the conditions of the theorem on the interval  $[-1,\mathrm{\hspace{0.17em}1}]$  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	Extremum
Related topic	LeastAndGreatestNumber
Related topic	FermatsTheoremStationaryPoints
Related topic	MinimalAndMaximalNumber
Defines	absolute minimum
Defines	absolute maximum