Find the critical points and say whether they are maxima, minima or saddle points I have this problem:
Find the critical points and say whether they are maxima, minima or saddle points
$$f(x,y)=x^2y(2−x−y)$$
My answer:
$f_x = xy (4-3 x-2 y) $
$f_y = -x^2 (-2+x+2 y)$     then 
$xy (4-3 x-2 y)=0 , -x^2 (-2+x+2 y)=0$ the solutions are:
1)x=0 and $y \in{} \mathbb{R}$
2)x=1 and y=1/2
3)x=2 and y=0
I have three critical points, 
$f_{xx} = -2 y (-2+3 x+y) $
$f_{yy} = -2 x^2 $
$f_{xy} = x (4-3 x-4 y) $, using   
$D=f_{xx}f_{yy}-(f_{xy})^2 = -16 x^2+24 x^3-9 x^4+24 x^2 y-12 x^3 y-12 x^2 y^2$
x=1 , y=1/2 then D=0 but $f_{xx}=-2y (-2+3 x+y)= -2/3$ concluding (1,1/2) is a maxima point
x=2 , y=0 then D=-16 we say that (2,0) is a saddle points
x=0 , $y \in{} \mathbb{R}$ then D=0 the point  (x,a) with $a \in{} \mathbb{R}$ is a saddle points because for example if x=0 , a=-8 $fxx=-160$ (maxima) but if x=0 , a=1 then $f_{xx}=2$ (minima)
Is my exercise correct?

Look at the Hessian matrix and consider its eigenvalues. If the determinant is negative, then you get a saddle point. If the determinant is zero, the test is inconclusive; if the determinant is positive and $f_{xx} > 0$, then you get a minima; and if the determinant is positive and $f_{xx} < 0$ , then you have a maxima.
Now, can you see whether you are correct?
Hessian matrix is just the matrix
\begin{pmatrix}
f_{xx} & f_{xy} \\
f_{yx} & f_{yy} \\
\end{pmatrix}
