Critical points of $f(x,y) = x^{3}y^{2} -x^{4}y^{2} - x^{3}y^{3}$ What are the critical points of $f(x,y)= x^{3}y^{2} -x^{4}y^{2} - x^{3}y^{3} ?$
Clearly $f_{x} = 3x^{2}y^{2} - 4x^{3}y^{2} -3x^{2}y^{3}$ $(1)$
And $f_{y} = 2x^{3}y -2x^{4}y -3x^{3}y^{2}$.   $(2)$
Solving these equation, the critical point I'm getting is $(1/2,1/3)$
Now, is there any other critical points$?$ 
From equation $(1)$, $x=0$ also satisfy this equation, but by putting $x=0$ in equation $(2)$ we get nothing.So there should be no other critical points. 
But answer also inculde $(0,0)$ with two more critical points other than $(1/2,1/3)$.
What is exact procedure of finding critical points for functions of two variables$?$
 A: Given the function $f : \mathbb{R}^2 \to \mathbb{R}$ of law
$$
f(x,\,y) := x^3\,y^2\left(1 - x - y\right),
$$
your critical points are all those that cancel the gradient:
$$
\nabla f(x,\,y) = (0,\,0) 
\; \; \; \Leftrightarrow \; \; \;
\begin{cases}
x^2\,y^2\left(3 - 4\,x - 3\,y\right) = 0 \\
x^3\,y\left(2 - 2\,x - 3\,y\right) = 0
\end{cases}
$$
from which it follows that $f$ presents an isolated critical point:
$$
(x,\,y) = \left(\frac{1}{2},\,\frac{1}{3}\right)
$$
and two lines of critical points (therefore not isolated):
$$
(x,\,y) = (0,\,t)\,, \; \; \; (x,\,y) = (t,\,0)\,, \; \; \; \text{with} \; t \in \mathbb{R}\,.
$$

Now, calculated the Hessian matrix for $f$:
$$
H_f(x,\,y) =
\begin{bmatrix}
6\,x\,y^2\left(1-2\,x-y\right) & x^2\,y\left(6-8\,x-9\,y\right) \\
x^2\,y\left(6-8\,x-9\,y\right) & 2\,x^3\left(1-x-3\,y\right)
\end{bmatrix}
$$
the nature of the isolated critical point is soon determined:
$$
\det H_f\left(\frac{1}{2},\,\frac{1}{3}\right) = \frac{1}{144} > 0 \; \; \land \; \; H_f\left(\frac{1}{2},\,\frac{1}{3}\right)_{1,1} = -\frac{1}{9} < 0
$$
from which it's deduced that $\left(\frac{1}{2},\,\frac{1}{3}\right)$ is a local maximum point for $f$.

Since the non-isolated critical points always cancel the Hessian determinant, this study is inconclusive. For this reason, for these other critical points it's better to study the sign of $\small f(x,\,y) - f(0,\,t)$ and $\small f(x,\,y) - f(t,\,0)$, which specifically is equivalent to studying the sign of $f$:

from which it's easy to deduce that:


*

*$(0,\,t)$ with $t \in \mathbb{R}$ it's a line of points neither of minimum nor of local maximum for $f$;

*$(t,\,0)$ with $t < 0 \, \vee \, t > 1$ are two half lines of local maximum points for $f$;

*$(t,\,0)$ with $0 < t < 1$ it's a segment of local minimum points for $f$;

*$(0,\,0), \; (1,\,0)$ are two points neither of minimum nor of local maximum for $f$.

