1
$\begingroup$

I never had seen this exercise, but I'm confused again, I don't know what I have to use.

I have the surface $S=\{(x,y,z)\in \mathbb{R}^3|xy+xz+yz=1,x>0,y>0,z>0\}$, is $S$ regular?. Then, if $S$ is regular, I have to found the higher value of $f$ defined by $f(x,y,z)=xyz$ where $(x,y,z)$ is in $S$.

Can anyone help me?

  • 0
    There are three projections $S \to \mathbb R^2$, given by $$(x,y,z)\mapsto\begin{cases}(x,y) \\ (x,z) \\ (y,z).\end{cases}$$ For each of these maps, figure out where it is regular. If (for every point $p \in S$, at least one of these three maps is regular at $p$), then $S$ is regular.2012-11-10

2 Answers 2