Minimization on compact region I need to solve the minimization problem
$$\begin{matrix} \min & x^2 + 2y^2 + 3z^2 \\ subject\;to & x^2 + y^2 + z^2 =1\\ \; & x+y+z=0
\end{matrix}$$
I was trying to verify the first order conditions using lagrange multipliers
$$\left\{\begin{matrix}
2x = 2\lambda_1x + \lambda_2 \\
4y = 2 \lambda_1 y + \lambda_2 \\
6z = 2\lambda_1 z + \lambda_2
\end{matrix}\right.$$
Summing the 3 equations we get
$$ 2y + 4z = 3 \lambda_2$$
Multiplying the first one for x, the second for $y$ and third for $z$ and summing:
$$ 2 + 2y^2 + 4z^2 = 2 \lambda_1$$
I just got stuck there and I have no idea how to proceed. We can check the answer here  but I couldn't find none of the variables. Any idea?
Thanks in advance!

since you have
$$x=\dfrac{\lambda_{2}}{2(1-\lambda_{1})},y=\dfrac{\lambda_{2}}{4-2\lambda_{1}},z=\dfrac{\lambda_{2}}{6-2\lambda_{1}}$$
and $x+y+z=0$ so we have
$$\dfrac{\lambda_{2}}{2}\left(\dfrac{1}{1-\lambda_{1}}+\dfrac{1}{2-\lambda_{1}}+\dfrac{1}{3-\lambda_{1}}\right)=0$$
so we have$\lambda_{2}=0$ or
$$\dfrac{(2-\lambda_{1})(3-\lambda_{1})+(1-\lambda_{1})(3-\lambda_{1})+(1-\lambda_{1})(2-\lambda_{1})}{(1-\lambda_{1})(2-\lambda_{2})(3-\lambda_{3})}=0$$
then we have 
$$3\lambda^2_{1}-12\lambda_{1}+11=0\Longrightarrow \lambda_{1}=2\pm\dfrac{1}{\sqrt{3}}$$
then you can easy to find this respect $\lambda_{2}$
