It is absurd to try solving this using Lagrange Multiplers since the minimum is obviously -2 and the maximum is +2 and because of the periodic nature of f(x,y) these can be found by conventianal calculus techniques.

If you decide to go down the route of Lagrange Multipliers you first have to look for the calculus like extrema without constraints. In which case you will find the extrem values mentioned above, and as at least one point exists in the feasible region for each, and they are obviously global maximum and mininum you are done before introducing any multipliers.

CB