Find the critical points of the function $\displaystyle f(x,y)=x^3+6xy+y^3$ determine which are local minima, which are local maxima, and which are saddle points.
You need to set the first order partial derivatives equal to zero.
That creates 'flat spots' (critical points) that may be mins or maxs.
THEN via the second order partial derivaives we can determine what exactly those candidates are.
$\displaystyle f_x=3x^2+6y$ and $\displaystyle f_y=3y^2+6x$.
YOU need these BOTH to be zero.
FIND all those points and then do the second derivative test.