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.
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.) $\displaystyle y=\frac{-1}{x}$. Then plug this in for y in the second equation and you get $\displaystyle 0=x-\frac{4}{x}$. Solve this equation for x and you get $\displaystyle x=\pm2$. Plugging in to (1.), you get $\displaystyle y(2)=\frac{-1}{2}$ and $\displaystyle y(-2)=\frac{1}{2}$.
When the system involves partial derivatives it seems to make people hesitate, but you can essentially just treat $\displaystyle f_{x}=0$ and $\displaystyle f_{y}=0$ 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.