<s>[INST] Find minimum and maximum of $P=a+b+c$ Let $a,b,c\ge 0$ such that $a^2+b^2+c^2+abc=4$. Find minimum and maximum of $$P=a+b+c$$

+)Maximum: Let $x=\frac{2\sqrt{ab}}{\sqrt{\left(c+a\right)\left(c+b\right)}};y=\frac{2\sqrt{bc}}{\sqrt{\left(a+b\right)\left(a+c\right)}};z=\frac{2\sqrt{ca}}{\sqrt{\left(b+c\right)\left(b+a\right)}}$
And by AM-GM inequality i proved that $P\le 3$ when $a=b=c=1$
+)Minimum: I will prove $P\ge 2$ and the equality occurs when $a=b=0;c=2$ but failed. I tried substituting and uvw 
 [/INST] For the minimum, note $P^2 = 4+2(ab+bc+ca)-abc \geqslant 4+6(abc)^{2/3}-abc \geqslant 4$, as $abc \in [0, 1]$ from the obvious AM-GM $4 = a^2+b^2+c^2+abc \geqslant 4(abc)^{3/4}$ and $f(t)=6t^2-t^3$ is increasing in $t\in [0, 1]$. 
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