Maximum value of $a+b+c$ in an inequality Given that $a$, $b$ and $c$ are real positive numbers, find the maximum possible value of $a+b+c$, if
$$a^2+b^2+c^2+ab+ac+bc\le1.$$
From the AM-GM theorem, I have
$$a^2+b^2+c^2+ab+ac+bc\geq 6\sqrt[6]{a^4b^4c^4} = 6\sqrt[3]{a^2b^2c^2} \\
6\sqrt[3]{a^2b^2c^2} \le1 \\
a^2b^2c^2 \le \frac{1}{216} \\
abc \le \frac{\sqrt{6}}{36}$$
However, I don't know where to go from here.

$$a+b+c=\sqrt{\sum_{cyc}(a^2+2ab)}\leq\sqrt{\sum_{cyc}\left(a^2+\frac{1}{2}a^2+\frac{3}{2}ab\right)}\leq\sqrt{\frac{3}{2}}.$$
The equality occurs for $a=b=c$ and $\sum\limits_{cyc}(a^2+ab)=1,$ which says that we got a maximal value.
