<s>[INST] 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!
 [/INST] An elementary approach: using $2xy\leq x^2+y^2$, you can write
$$
xy=\frac{1}{3}(2xy+xy)\le\frac{1}{3}(x^2+y^2+xy)=\frac{1}{3}1=\frac{1}{3}
$$
with equality iff $x=y=\frac{1}{\sqrt{3}}$. For minimum, note that
$$
xy+1=xy+(x^2+y^2+xy)=(x+y)^2\geq 0\implies xy\geq-1
$$
where equality realizes when $(x,y)=(1,-1)$ or $(x,y)=(-1,1)$.
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