I'm stuck doing this problem.
Optimize $f(x,y,z) = xyz$ restricted to $g(x,y,z) = x^2+2y^2+3z^2 = 6$
First, I found ${\nabla}f$ and ${\lambda}{\nabla}g$, and for Lagrange Multipliers, I got these four equations
$$ \begin{matrix} yz = 2{\lambda}x \hspace{8mm} (1) \\ xz = 4{\lambda}y \hspace{8mm} (2) \\ xy = 6{\lambda}z \hspace{8mm} (3) \\ x^2+2y^2+3z^2 = 6 \hspace{8mm} (4) \end{matrix} $$
But I don't know how to find $x,y,z$ or $\lambda$. I've been trying by summing two of those equiations, but from the expressions I got (which are too complicated to write here) I don't know what to find first.
Thanks for your help
$\textbf{Hint:}$ First check that $\lambda \neq 0$. Next multiply (1) by $x$ and (2) by $y$, you get:
$$xyz=2\lambda x^2$$
$$xyz=4 \lambda y^2$$
So $2 \lambda x^2=4 \lambda y^2$, then $x^2=2y^2$. The same with (1) and (3), next (2) and (3), finally substitute to (4).