已知x>0,y>0,且x+y+3=xy,则x+2y的最小值

Question

已知x>0,y>0,且x+y+3=xy,则x+2y的最小值

Answer 1

x+y+3=xy
(y-1)x=y+3
x>0 y>0 y+3>0,要等式成立,只有y-1>0
x=(y+3)/(y-1)=(y-1+4)/(y-1)=1+ 4/(y-1)
x+2y
=1+4/(y-1) +2(y-1) +2
=2(y-1) +4/(y-1) +3
由均值不等式得2(y-1)+4/(y-1)≥2√8=4√2,此时x+2y有最小值4√2
2(y-1)=4/(y-1)
(y-1)²=2
y-1>0 y-1=√2
y=√2+1
x+2y的最小值为4√2