Extremum of a function under constraints

Question

I have a function $f : E \subset R^n \to R$.

$E$ is compact and $f$ is continuous so the extremums exist.

But $E$ is not defined by an equation but an inequality, so i can't use the Lagrange method ... How do i do ?

Exemple : $n=2$, $E=$closed disc ($x^2+y^2\leq 1$) and $f(x,y)=(x+y)/(1+x^2+y^2)$

Thank you

Answer 1

Find the local extrema in $R^n$ and discard the ones outside $E$. Then find the extrema on the boundary of $E$. Finally pick the largest/smallest of them.