# Critical Points

• Dec 15th 2009, 08:42 AM
Volcanicrain
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.
• Dec 15th 2009, 09:08 AM
tonio
Quote:

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
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