Max and minimum height on the surface

Question

Determine the points of maximum and minimum height with relation with the $z$-axis of the surface $\Sigma=\{(u,v,w)\in \mathbb{R}^{3}; (u-2)^{2}+2(v+1)^{2}+(w-1)^{2}=1\}$. I know that the solution method relies on Lagrange multipliers, but my difficulty is finding the restriction.

Answer 1

Yes you can apply Lagrange multipliers to

• $f(u,v,w)=w$

with constraint

• $(u-2)^{2}+2(v+1)^{2}+(w-1)^{2}=1$