<s>[INST] Resolve this system: Im tried to resolve this problem:
$$\max\quad f\left( x,y \right) =xy\quad \text{s.a}\quad \begin{cases} x^2 +y^2+z^2 -1=0 \\ x+y+z=0 \end{cases}$$
Well, i form the lagrangian and the respective gradient, so i had this system to resolve:
$$\begin{cases} y+2\lambda_1 x+\lambda_2=0 \\ x+2\lambda_1 y+ \lambda_2=0 \\ x^2+y^2+z^2 -1=0 \\ x+y+z=0 \end{cases}$$
And i can't find all the solutions, i need help-
 [/INST] $z=-(x+y), x^2+y^2+(x+y)^2=1 \implies x^2+y^2+xy=\dfrac{1}{2},x^2+y^2\ge 2xy \implies 3xy \le \dfrac{1}{2}  \iff xy\le \dfrac{1}{6}$,
when $x=y \cap x^2=\dfrac{1}{6}$
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