$[15]$ Find the critical point$($s$)$ of $f(x,y)=-x^2-2xy\frac{1}{3}{y}^3-3y$ then use the Second Partials Test to classify each as a local maximum, local minimum, or saddle point. $($If the test is inconclusive, state that instead.$)$

Second Partials Test: Let $f$ have continuous second partial derivatives on an open region containing a point $(a,b)$ for which $f(a,b)=0$ and $f(a,b)=0.$ To test for relative extrema, consider the quantity

$d=f_{ar}(a,b)f_{yr}(a,b)-[f_{or}(a,b){]}^2$

If $d>0$ and $f_{}(a,b)>0,$ then $f$ has a relative minimum at $(a,b).$

If $d>0$ and then $f$ has a relative maximum at $(a,b).$

If then $(a,b,f(a,b))$ is a saddle point.

If $d=0$ the test is incoeslusive.