91Ó°ÊÓ

A function $f$of two or three variables has a global minimum or a global maximum at a point $P$.

(a) A function $f(x,y)$has a global or absolute maximum at $\left(x_0,y_0\right)$if $f\left(x_0,y_0\right)â‰\yen f(x,y)$for every point $(x,y)$in the domain of $f$.

(b) A function $f(x,y)$has a global or absolute minimum at $\left(x_0,y_0\right)$if $f\left(x_0,y_0\right)≤f(x,y)$for every point $(x,y)$in the domain of $f$.

Examples:

(1) The function, localid="1654246914302" $f(x,y)=x^2+y^2$has global minimum at localid="1654246955750" $(0,0)$as $f(0,0)=0$and for any other real values of $(x,y)$, $f$will always be greater than zero. That is, $f\left(0,0\right)≤f(x,y)$.

(2) The function, localid="1654247685007" $f(x,y)=1-x^2-y^2$has global maximum at localid="1654247708452" $\left(0,0\right)$as $f(0,0)=1$and for any other real values of $(x,y)$, $f$will always be lesser than $1$. That is, role="math" localid="1654248014328" $f(0,0)â‰\yen f(x,y)$.