# Find the critical points.

• Sep 27th 2009, 09:13 PM
jebckr
Find the critical points.
f(x,y)=xy+lnx+2y^2

I know fx=y+(1/x) and fy=x+4y.

So my question is how do I find the critical points? I know it involves setting the two equations equal to 0, but I'm having a brain lapse. Please help.
• Sep 27th 2009, 09:25 PM
davesface
As you said, set both equations equal to 0. Then take one of the equations and solve it for a variable. For instance, you could say (1.) $y=\frac{-1}{x}$. Then plug this in for y in the second equation and you get $0=x-\frac{4}{x}$. Solve this equation for x and you get $x=\pm2$. Plugging in to (1.), you get $y(2)=\frac{-1}{2}$ and $y(-2)=\frac{1}{2}$.

When the system involves partial derivatives it seems to make people hesitate, but you can essentially just treat $f_{x}=0$ and $f_{y}=0$ like you would any other set of two equations involving two independent variables. It's just the interpretation of the results that makes the system special.
• Sep 27th 2009, 09:38 PM
jebckr
Thanks, this really cleared it up for me.