I'm in cal III right now and I believe the easiest way to go about a problem specifically like the example you've shown is to first use the partials with respect to x and y to find any critical points. Obviously with this method you don't find any because the function is that of a plane. Now you need to test the constraint function. If you think about the constraint as a function on the xy-plane you see that you have a circle. This will produce a minimum and a maximum on the plane given that the plane is not horizontal which in this situation it isn't. From here you can change the constraint function to polar coordinates. Once you have done this the objective function becomes a function of one variable, i.e. theta. From here take the derivative of the function of theta and solve for theta given the derivative equals 0. This will give you two critical points which you can then test in your original objective function to determine which is a maximum and which is a minimum.

I hope that made sense. I'm on mobile right now so it's hard to type all the work out but I believe this is the best way to go about this. We just finished this section so there might be an easier way to do it that I'm unaware of.