KKT conditions

Consider the problem
min xy
s.t. x^2 + y^2 <=2
x + y >=0
Show that the KKT conditions are satisfied at every local minimizer.
Compute all the KKT points, and show that strict complementarity is not
satisfied at any KKT point. Use second order necessary/sufficient
conditions to determine which KKT point(s) is a local minimizer.