Stationary points with 2 variables

Hi,

Locate and classify all of the stationary points of the function:

$\displaystyle f(x,y) = e^x(3x - y^3 + 3y)$.

I have got as far as doing the partial derivatives for the terms but I am unsure if the partial derivatives I got are right

What I got for the derivatives so far are:

$\displaystyle f_{x} = 3e^x - e^xy^3$

$\displaystyle f_{xx} = 3e^x - e^xy^3$ <-- unsure on this one

$\displaystyle f_{y} = 3xe^x - 3e^xy^2 + 3$

$\displaystyle f_{yy} = 3xe^x - 6e^xy$

$\displaystyle f_{xy} = 3e^x - e^xy^3$

I am sure that after I have found these I am meant to equate them to zero (because they are stationary points) and then find x and y and then put the in one of the equations (I think its the original one if I remember correctly) and get the coordinates of the stationary points.