Finding local max, min and saddle points

[maff is tuff]

Homework Statement

f(x,y)=(1+xy)(x+y)

Homework Equations

The Attempt at a Solution

I started out by expanding and got:

$x+y+x^2y+xy^2$

Then I found all my partial derivatives and second derivatives:

$f_{x}=1+2xy+y^2, f_{y}=1+2xy+x^2, f_{xx}=2y, f_{yy}=2x, f_{xy}=2(x+y)$

I know that both first partial derivatives must equal zero so I get:

$f_{x}=1+2xy+y^2=0$ and $f_{y}=1+2xy+x^2=0$

This is the part I am stuck at; I can't find the critical points. I notice that there is symmetry so I tried subtracting the equations but I got y=x and got:

$f_{x}=1+2x(x)+(x)^2=1+2x^2+x^2=0=1+3x^2=0---->x^2=-\frac{1}{3}$

I also tried setting $f_{x}$ and $f_{y}$ equal to each other but that didn't seem to work.

Thanks in advance for the help

 

[maff is tuff]

I think I may have got it. When I got that $x^2=y^2$ I didn't account for that y could equal (-x). I did that and it got the right answer.