<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] Obviously, $a,b,c\in \left[0,2\right]$ so $abc\leqslant 8$. Thus
$$abc = \sqrt[3]{abc}\cdot \sqrt[3]{(ab)(bc)(ca)} \leqslant 2\left(\frac{ab+bc+ca}{3}\right) \leqslant 2(ab+bc+ca).\tag{1}$$
Thus
$$ 4=a^2+b^2+c^2+abc \leqslant (a+b+c)^2.$$
Equality occurs when we have equality in (1), which happens only when both sides are $0$.
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