本帖最后由 facebooker 于 2019-10-8 20:16 编辑
回复 9# kuing
$
\sqrt{x^2+8}\geqslant \frac{x+5}{2}+\frac{4}{3}(x-1)^2
\\\sqrt{y^2+8}\geqslant \frac{y+5}{2}+\frac{4}{3}(y-1)^2
\\ \Rightarrow \sqrt {x^2+8}+\sqrt{y^2+8}\geqslant6+\frac{4}{3}({(x-1)^2+(y-1)^2})=6+
\frac{4}{3}({(x-1)^2+(y-1)^2})=2(1-xy)
\\ \Rightarrow \sqrt {x^2+8}+\sqrt{y^2+8}\geqslant6+\frac{4}{3}({(x-1)^2+(y-1)^2})+\sqrt{xy+8}\geqslant 6+ \frac{8}{3}(1-xy)+\sqrt{xy+8}\geqslant 9,
\\ \Rightarrow \frac{8}{3}(1-xy)+\sqrt{xy+8}\geqslant 3,0<xy\leqslant 1.
$ |