Difference between revisions of "Extrema"

Latest revision as of 02:12, 15 February 2008

The upper and lower bounds of a real valued function are of interest in several situations in pure as well as applied Mathematics

Let $A$ be a set

Let $f:A\rightarrow\mathbb{R}$

Let the set $f(A)$ be bounded

Then $M=\sup\{f(A)\}$ is called the Absolute or Global maximum of $f$

and $m=\inf\{f(A)\}$ is called the Absolute or Global minimum of $f$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $c\in [a,b]$

$f(c)$ is said to be a Local maximum iff $\exists\delta>0$ such that $f(c)=\sup\{f(V_{\delta}(c))\}$

$f(c)$ is said to be a Local minimum iff $\exists\delta>0$ such that $f(c)=\inf\{f(V_{\delta}(c))\}$

@@ Line 4: / Line 4: @@
 Let <math>A</math> be a set
-Let <math>f:A\rightarow\mathbb{R}</math>
+Let <math>f:A\rightarrow\mathbb{R}</math>
 Let the set <math>f(A)</math> be bounded
@@ Line 20: / Line 20: @@
 <math>f(c)</math> is said to be a '''Local minimum''' iff <math>\exists\delta>0</math> such that <math>f(c)=\inf\{f(V_{\delta}(c))\}</math>
+==See Also==
+*[[Function]]
+*[[Neighbourhoods]]
+*[[Derivative]]
+{{stub}}