There are two parts to this.

Case 1 is x^2 + y^2 = 4, i.e., the boundary.

Here f=x^2 + y^2 +y^2=4+y^2.

So get the max/min of g(y)=4+y^2 where

By inspection the max occurs at y=+-2 and min is at y=0 (on the boundary).

Case 2, the interior of this circle.

you can get the global max by calc 3, but clearly the function is increasing as you move away from the origin.

So the max's occur on the boundary, see case 1.

And the min of a sum of squares is zero at the origin.

So no calculus is even needed here.