For each problem, locate the critical points and classify each one using the second derivative test.

a. \(f(x, y)=(x+y)^{2}\).

b. \(f(x, y)=x^{2} y+x y^{2}\).

c. \(f(x, y)=x^{4} y+x y^{4}-x y\).

d. \(f(x, y)=x^{2}-3 x y+2 x+10 y+6 y^{2}+12\).

e. \(f(x, y)=\left(x^{2}-y^{2}\right) e^{-y}\).