Finding the minimum on subset of $R^3$ of the function $J(x,y,z)=x^2+y^2+z^2$

Question

Let $$J(x,y,z)=x^2+y^2+z^2$$ and the set $$C=\{(x,y,z)|(x-1)^4+(y-2)^2 +z\le 0\}$$

How I can show by simple geometrical reasoning that the minimum on $C$ is reached on $$D = \{(x,y,z)|(x-1)^4+(y-2)^2 +z=0\}$$.

Thanks!

Answer 1

The gradient of $J$ is not equal to $0$ on the interior of $C$, therefore there are no local minima on the interior of $C$. Hence the minimum in $C$ must be attained on the boundary, i.e. $D$.