Maximum and minimum of $f(x,y)=xy$ when $x^2 + y^2 + xy =1$ It is asked to find the maximum and minimum points of the function
$$f(x,y)=xy$$
when $x^2 + y^2 + xy=1$
I've tried Lagrange and obtained
$$\lambda = \frac{y}{2x+y}=\frac{x}{2y+x}$$
but what should I do with this? Any other suggestion?
Thanks!

you can do this without calculus. the $x$-coordinates of the points common to both $xy = k$ and $x^2 + y^2 + xy = 1$ satsfies $ x^2 + k^2/x^2 + k - 1 = 0$ which can be turned into a quadratic equation for $u = x^2$ as $$u^2 + (k-1)u^2 + k^2 = 0$$ whose discriminant $ 1 - 2k - 3k^2 $ is positive for $-1 \le k \le 1/3.$ 
so the maximum value of $f$ is $2/3$ and the minimum of $f$ is $-1.$  
