$(15$ points$)$ What are the critical points of the function $f(x,y)=x^{2}y+x{y}^2-xy?$

$(x,y)=(0,0)$

$(x,y)=(0,0),(x,y)=(1,0)$ and $(x,y)=(0,1)$

The function has no critical points

$(x,y)=(0,0),(x,y)=(1,0),(x,y)=(0,1)$ and $(x,y)=(\frac{1}{3},-\frac{1}{3})$

None of the above