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:

[itex]x+y+x^2y+xy^2[/itex]

Then I found all my partial derivatives and second derivatives:

[itex]f_{x}=1+2xy+y^2, f_{y}=1+2xy+x^2, f_{xx}=2y, f_{yy}=2x, f_{xy}=2(x+y)[/itex]

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

[itex]f_{x}=1+2xy+y^2=0[/itex] and [itex]f_{y}=1+2xy+x^2=0[/itex]

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:

[itex]f_{x}=1+2x(x)+(x)^2=1+2x^2+x^2=0=1+3x^2=0---->x^2=-\frac{1}{3}[/itex]

I also tried setting [itex]f_{x}[/itex] and [itex]f_{y}[/itex] 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 [itex]x^2=y^2[/itex] I didn't account for that y could equal (-x). I did that and it got the right answer.