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 rightsided derivative in $a$ or the leftsided 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 onesided derivatives, gains its least value in the endpoint $x=1$ 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  20130322 15:38:57 
Last modified on  20130322 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 