We find the global extrema of $f(x,y)=2x^2 + y^2$ on the region $x^2 + y^2 <= 4$.
Interior Critical Points
Boundary $x^2 + y^2 = 4$ parametrized with $r:t\in[0,2\pi]\to\mathbb{R}^2$ defined by $r(t) = (2\cos(t), y = 2\sin(t))$.
Global minimum of 0 at the critical point (0, 0).
Global maximum of 8 at (2, 0) and (-2, 0).