1035

已知 ${\displaystyle x,y>0}$, ${\displaystyle (x+y+xy)(x+y-xy)=xy}$. 求 ${\displaystyle x+y+xy}$ 和 ${\displaystyle x+y-xy}$ 的最小值.

简记 ${\displaystyle A=x+y,B=xy}$, 则 ${\displaystyle A^2\ge 4B,A^2=B^2+B}$, 联立后解得 ${\displaystyle B\ge 3}$ (等号成立当且仅当 ${\displaystyle x=y=\sqrt{3}}$). 则$$\begin{array}{rl}& x+y+xy=A+B\ge 2\sqrt{B}+B\ge 3+2\sqrt{3},\\ & x+y-xy=A-B=\sqrt{B^2+B}-B=\frac{1}{1+\sqrt{1+\frac{1}{B}}}\ge \frac{1}{1+\sqrt{1+\frac{1}{3}}}=2\sqrt{3}-3.\end{array}$$