<s>[INST] Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$. 
Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.

I tried many method, such as AM-GM, but all of them failed.
Thank you.
 [/INST] use this 
$$(x+y+z)^2+3(x+y+z)\ge x^2+y^2+z^2+x+2y+3z$$
and let $x+y+z=t$,
then $$t^2+3t\ge \dfrac{13}{4}$$
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