(* We find the global extrema of 2 x^2+y^2  on the region x^2+y^2≤4 *)

f=2x^2+y^2
Plot3D[f,{x,-2,2},{y,-2,2},RegionFunction->Function[{x,y},x^2+y^2≤4]]
ContourPlot[f,{x,-2,2},{y,-2,2},RegionFunction->Function[{x,y},x^2+y^2≤4]]

Gf=Grad[f,{x,y}]
VectorPlot[Gf,{x,-2,2},{y,-2,2},RegionFunction->Function[{x,y},x^2+y^2≤4]]

(* Critical points *)
crits=Solve[Gf=={0,0},{x,y}]

H=D[Gf,{{x,y}}] /. crits[[1]];
MatrixForm[H]
Det[H]
(* The only critical point inside the region is a local minimum.  *)
f/.{x->0,y->0}

(* Boundary *)
g=f/.{x->2 Cos[t],y->2 Sin[t]}
Plot[g,{t,0,2 Pi}]
gcrits=Solve[{D[g,t]==0,0<=t <=2 Pi},t ]
g /. gcrits[[1]]
g /. gcrits[[2]]
g /. gcrits[[3]]
g /. gcrits[[4]]

(* Global ninimum of 0 at the critical point (0,0) *)
(* Global maximum of 8 at two point on the boundary *)
{2 Cos[t],2Sin[t]} /. gcrits[[1]]
{2 Cos[t],2Sin[t]} /. gcrits[[3]]