I have a set $S=\{(x,y,z) \in \mathbb{R}^3 \mid g(x,y,z)=x^2+y^2-z^2 = 1\}$ on which we have a function $f:S \rightarrow R$ defined by $$ f(x,y,z)=x+y+z^2 $$ I have to find all global minima and maxima. By Lagrange, I have found the global minima to be at $(x=-\sqrt{\frac{1}{2}}, y=-\sqrt{\frac{1}{2}}, z=0)$. If $z \neq 0,$ I get a complex solution, which is not what we are looking for.
Before solving I have to argue that such a minimum/maximum exist or does not exist. I know that a continious function on a compact set assume its minima and maxima, but $S$ is unbounded in our case and therefore not compact. Talking about maxima I have shown that a maximum cannot exist by contradiction. Assuming that there exists a maximum with some value, and then choosing another point based on this value, that satisfies $g=1,$ which turns out to have a bigger value according to $f.$
How do I argue that the minima does exist and is there an easier way to show that there are no global maximum? I guess I have to show that $f$ is closed and bounded from below, subject to the condition, because then a minima will exist, but I cant figure out how. I know from the Hessian that $f$ is convex while $g$ is not convex, can I use that in any way?
For existence of minima, you have to show that $f(x,y,z)\to\infty$ if $\|(x,y,z)\|\to\infty$. Then sets of the type $\{(x,y,z):f(x,y,z)\le C\}$ are bounded.
Let $(x,y,z)$ satisfy the constraint. Then $z^2=x^2+y^2-1$ and $$ f(x,y,z)= x+y+z^2 = x+y+ \frac12 z^2 +\frac12 x^2 + \frac 12 y^2-\frac12 =\frac12(x+1)^2 + \frac12(y+1)^2 +\frac12 z^2 - \frac32. $$ The right hand-side tends to $+\infty$ for $\|(x,y,z)\|\to\infty$.
To see that this is enough, take the feasible point $(1,0,0)$. Any minimum has to have a smaller value of $f$ than $1$. So we can restrict our search to all feasible points with $f\le 1$. Due to the above inequality, such $(x,y,z)$ satisfy $$ \frac12(x+1)^2 + \frac12(y+1)^2 +\frac12 z^2 - \frac32 \le 1 $$ or $$ (x+1)^2 + (y+1)^2 + z^2 \le 5, $$ that is, they are in a circle around $(-1,-1,0)$ with radius $\sqrt 5$. The set of such feasible points is compact, so we get existence of minimum.