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.

You have to set:
$$F_x=F_y=0....(1)$$
Then solve the equations and get some $(x,y)$ witch holds $(1)$.
$\textbf{HINT:}$


*

*$(1,0), (0,1)$ are critical points.

*$x^2-4xy-x+y^2+y=0$

*$(x-y)^2-(x+y) -2xy=0$

