Question

已知正数x y满足xy=x+9y+7 则xy的最小值是

Answer 1

xy=x+9y+7.===>x(y-1)=9y+7.===>x=(9y+7)/(y-1).由x,y>0可知,y>1.且xy=[y(9y+7)]/(y-1).又y(9y+7)=[(y-1)+1][9(y-1)+16]=9(y-1)²+25(y-1)+16.故xy=25+9(y-1)+[16/(y-1)].再由y-1>0及均值不等式可得：xy=25+9(y-1)+[16/(y-1)]≥25+2√(9×16)=49.等号仅当y=7/3时取得,此时x=21.故(xy)min=49.