<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] With the maximal value you are right.
For $c=0$ we get a value $2$ and it'a a minimal value because
$$2\sum_{cyc}\sqrt{\frac{ab}{(a+c)(b+c)}}\geq2$$ it's
$$\sum_{cyc}\sqrt{ab(a+b)}\geq\sqrt{\prod_{cyc}(a+b)},$$ which is obvious after squaring of the both sides:
$$\sum_{cyc}(a^2b+a^2c+2\sqrt{a^2bc(a+b)(a+c)})\geq\sum_{cyc}(a^2b+a^2c+\frac{2}{3}abc).$$
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