Let $K=\{(x,y,z)\in\mathbb{R}^3|\,x^2+\frac{1}{2}y^2+2z^2\leq 1\}$. Let $F:\mathbb{R}^3\to\mathbb{R},\,(x,y,z)\mapsto x^2+e^{-y^2}+4z^2$. Find the global extrema of $F$ on K.
I've first examined the interior of $K$ and found out that $(0,0,0)$ is a critical point. After that, I've considered the boundary and tried to use Lagrangian multipliers by setting $\tilde{F}(x,y,z,\lambda):=F(x,y,z)-\lambda(x^2+\frac{1}{2}y^2+2z^2-1)$ and looking for extrema of $\tilde{F}$ but it seems as though that things get very complicated from there on and I haven't been able to find a solution so far.