# Critical points for multivariable functions

Suppose you have the function, $$\space f(x,y) =5xy(9-x-y)$$ and you have to find the critical points of $$\space f(x,y)$$. To start, I found the $$f_x$$ and $$f_y$$ and set each to $$0$$. Then, I tried to solve the system of equations:

$$f_{x} = -5y^2-10xy+45y = 0$$ $$f_y = -5x^2-10xy+45x = 0$$

At this point, is there a simple method to solve this system? I am thinking I should multiply $$\space f_y$$ by $$-1$$, yielding: $$-5y^2-10xy+45y = 0$$ $$5x^2+10xy-45x = 0$$ and then adding the equations together, but I don't think that is useful. Additionally, I think I can solve the top equation for $$x$$ and then plug the resulting equation into $$f_y$$, but that is tedious.

Thanks!

• $$(x=0\land y=9)\lor (x=3\land y=3)\lor (x=9\land y=0)\lor (y=0\land x=0)$$ – Moo Oct 17 '18 at 3:45
• @Moo Oh! How did you do that? – Art Oct 17 '18 at 3:47
• Solve the first equation for $x$ and substitute into the second. That gives you a quadratic equation. – Moo Oct 17 '18 at 3:54
• @Moo Ok. Thanks. – Art Oct 17 '18 at 4:10

I think your idea is useful because it gives you the equation (after cancelling 5) $$x^2-y^2-9(x-y)=0.$$ Now use $$x^2-y^2=(x+y)(x-y)$$ to factor out $$x-y$$ $$(x-y)(x+y-9)=0.$$