6.
$ P=(x+y)^{2}(\frac{1}{x^{2}+y^{2}}+\frac{1}{xy})$
$ =(x+y)^{2}\left(\frac{1}{x^{2}+y^{2}}+\frac{1}{2xy}+\frac{1}{2xy}\right)$
$\geq (x+y)^{2}\left(\frac{4}{(x+y)^{2}}+\frac{1}{\frac{(x+y)^{2}}{2}}\right)=6$
dấu "=" $\Leftrightarrow x=y$
9.
Ta có $(2\sqrt{x+1}+\sqrt{x-2})^2 \le (2^2+1^2)(2x-1)=5(2x-1) \overset{BDTD}{\le} (x+2)^2$
$\Rightarrow 2\sqrt{x+1}+\sqrt{x-2} -x\le 2$
$\Leftrightarrow \max B=2016$