Can anyone help with determining the nature of the critical points for the function:

f(x,y) = x^3 - x + y^2 - 2xy?

By using the first partial derivatives and setting them to zero, I determined the critical points to be f(1,1) and f(-1/3,-1/3).

To determine the nature of these critical points I calculated the second partial deratives and the discriminant to find that f(1,1) is a local minimum which appears to agree with the 3d plot for the function.

However my problem is at f(-1/3,-1/3), the determinant is less than 0 which implies a saddlepoint, but I believe that it should be a local maximum from looking a the plot.

Can anyone assist me is seeing the error of my ways? Any suggestions would be greatly appreciated.