1. ## Critical Points

Hi everyone!

I understand critical points very well, but I am just unsure... I'm doing a practice final with no answers, and there are a lot of points attributed to this question:

Find all critical points of f(x,y) = 3xy - x^2y - xy^2

I found (0,0) and (1,1) - are there others? Thank you very much.

2. Originally Posted by Volcanicrain
Hi everyone!

I understand critical points very well, but I am just unsure... I'm doing a practice final with no answers, and there are a lot of points attributed to this question:

Find all critical points of f(x,y) = 3xy - x^2y - xy^2

I found (0,0) and (1,1) - are there others? Thank you very much.

$\displaystyle f_x=3y-2xy-y^2=0 \Longrightarrow -y(y+2x-3)=0$

$\displaystyle f_y=3x-x^2-2xy=0 \Longrightarrow -x(x+2y-3)=0$

Solving the above system we get: $\displaystyle (0,0)\,,\,y+2x=3=x+2y\Longrightarrow y-x=0\Longrightarrow y=x\,\,\,and\,\,\,x+2x=3\Longrightarrow x=1$

Thus, indeed $\displaystyle (0,0)\,,\,(1,1)$ are the only function's critical points.

Tonio