Verified answer

f(x,y) = x³+y³−3xy+9

For critical points ∂f/∂x = 3x²−3y = 0 and ∂f/∂y = 3y²−3x = 0

Hence x(x³−1) = 0 → x=0 or x=1

The critical points are (0,0) and (1,1)

Applying the "2nd derivative test" we first find the 2nd derivatives.

(a test for pos/neg definite or indefinite H that only applies in 2 dimensions)

∂²f/∂²x = 6x, ∂²f/∂²y = 6y and ∂²f/∂x∂y = −3

Determinant of Hessian is det(H) = (6x)(6y) − (-3)(-3) = 36xy−9

For (0,0) : det(H)<0 so (0,0) is a saddle point

For (1,1) : det(H)>0 and ∂²f/∂²x>0 so (1,1) is a local minimum.