Finding the critical points of $f(x,y) = x y^2 - x^2 y + x y$ Trying to find the critical points of $f(x,y) = y^2x - yx^2 + xy$.
I took partial derivative with respect to x, so
$F_x = y^2 - 2xy + y$
$F_x = y(y - 2x + 1)$
Then with respect to y,
$F_y = 2xy - x^2 + x$
$F_y = x( 2y - x + 1 )$
From here I don't know how to find critical points. I've tried solving for $x$ or $y$, but I end up having to cancel out variables which would destroy potential answers. I've also tried setting $F_x = 0$ but four answers gets expanded to 8 answers and I don't know how to go from there. I have a feeling I'm missing something obvious because this is one of the early questions from the book.

Hint Recall that a point $(x_0, y_0)$ is a critical point if $F_x(x_0, y_0) = F_y(x_0, y_0) = 0$. Now,
$$F_x(x, y) = y (y - 2 x + 1),$$
and this is zero iff
$$y = 0 \qquad \text{or} \qquad y - 2x + 1 = 0.$$
Solving $F_y(x, y) = 0$ gives two similar conditions, giving in all four possible pairs of equations to solve.
