Absolute maximum and minimum, Multivariable

Find the absolute maximum and minimum:

f(x,y) = 3x2 + 2y, on set D = {(x,y) | x2 + y2 <= 2}

Within the boundary, there are no critical points:

Partial derivative with respect to y is 2, which is never zero.

If there are no critical points within the boundary, the extrema must occur on the boundary x2 + y2 = 2.

**I'm not quite sure how to calculate the extrema on the circle x2 + y2 = 2.** Here's what I've got, but I'm not sure how close I am to an answer:

x2 + y2 = 2, thus x2 = 2 - y2

So, substituting this into the original equation, f(x,y) = 3(2-y2) + 2y = 6-3y2+2y

We need to find the extrema on 6-3y2+2y, so we'll find the critical points:

-6y+2=0 when y = 1/3

I don't know what to do at this point, or if I've even done the right thing up until this point. I know there's another absolute max/min problem that was posted just before this one, but I know how to work such a problem. This one, on the other hand, is confusing me.