A function $$f\colon D \to \mathbb R$$ where $$(x,y,z) \in D \iff (x^2+y^2+4z^2-6 =0)$$ is given by this formula: $$f(x, y, z) = xyz$$ Prove - without finding them - that this function has a maximum and a minimum value.
I have thought about this problem and I came up with a possible:
We can either show that $D$ is a compact set, then $f$ will be uniformly continuous, and so $f$ will be continuous, and so - by the theorem of continuous functions - will have a minimum and a maximum value on $D$. However, it is not easy to prove that $D$ is compact.
Is there a better alternative to solving this problem, something a trifle less tedious?