Given $$f(x,y,z)=2x^2+y^2+z^2$$ I have to prove that f has a minimum on the set $$E=\{(x,y,z)\in \mathbb{R^3}: x^2yz=1\}.$$
$E$ is a closed set because of the fact that $g(x,y,z)=x^2yz$ is the preimage of the closed set $A=\{1\}$. But I cannot prove that it is bounded and use Weierstrass' theorem.
Is it right to use, instead, the multi variable Mean Value Theorem and show that there exists a point $\xi\in[x,y]\quad \forall x,y \in \mathbb{R^3}$ in which the function has a tangent plane?
$E$ is in fact not bounded. You should however be able to find $\tilde{E}\subset E$ such that $\tilde{E}$ is bounded and $f|_{E\setminus\tilde{E}}$ takes only "large" values.