<s>[INST] Find minimum value of $P = \frac{y + z}{x}$ 
Let $x,y,z > 0$ such that $3x(x+y+z)=yz$ . Find the minimum value of $$P=\frac{y+z}{x}$$


$$3x(x+y+z)=yz\Leftrightarrow 3(x+y)(x+z)=4yz$$
Or $$3(\frac{x}{y}+1)(\frac{x}{z}+1)=4$$
Let $a=x/y, b=x/z ( a,b>0)$
We have: $$4=3(a+1)(b+1)\ge3 \cdot 2 \sqrt  a \cdot 2 \sqrt  b=12 \sqrt {ab} \rightarrow \sqrt {ab}\le \frac 1 3$$
And $$P=\frac{y+z}{x}=\frac{1}{a}+\frac{1}{b}\ge \frac{2}{\sqrt {ab}}\ge 6$$
But by WA $Min_P=6+4\sqrt 3$. Is my solution is wrong?
 [/INST] Use Lagrange multipliers. Call the function you want to minimize $f$ and the constraint $g$. From the condition on the partial derivatives on $y$ and $z$ of $f - \Lambda g$ you can deduce that $y=z$. Finding $x$ with respect to $y$ gives you $x=\frac{\sqrt{12}-3}{3}y$, and this leads to the solution.
 You'd have to check this is actually a minimum (in principle this is only a critical point), but you should be able to verify that by yourself.
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