<s>[INST] 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!
 [/INST] $x^2+2y^2+3z^2=x^2+y^2+z^2+y^2+2z^2=1+y^2+2z^2, x=-(y+z),(y+z)^2+y^2+z^2=1 \implies y^2+yz+z^2=\dfrac{1}{2}$, now the problem become : find min of $y^2+2z^2$ with $y^2+yz+z^2=\dfrac{1}{2}$,it shoud be easier now.
or we can go further :$y=\dfrac{-z \pm \sqrt{2-3z^2}}{2},f(z)=\left(\dfrac{-z \pm \sqrt{2-3z^2}}{2}\right)^2+2z^2$ which is single variable function. 
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