Question

已知正数x，y满足：x+4y=xy，则x+y的最小值为

Answer 1

已知正数x，y满足：x+4y=xy，
1/y+4/x=1
x+y=(x+y)(1/y+4/x)
=x/y+4+1+4y/x
>=5+2根号下(x/yx4y/x)
=5+4
=9
x+y的最小值为9

Answer 2

∵x+4y=xy
∴(xy-x)=4y
∴x(y-1)=4y
∴x=4y/(y-1)＞0
又：y＞0
∴y-1＞0
∴y＞1
x+y = 4y/(y-1)+y = (4y-4+4)/(y-1)+y = 4 + 4/(y-1) + y
= 5 + 4/(y-1) + (y-1)
= 5 + {2/√(y-1)-√(y-1)}² + 4
= {2/√(y-1)-√(y-1)}² + 9 ≥9
最小值9

Answer 3

∵x+4y=xy
∴(xy-x)=4y
∴x(y-1)=4y
∴x=4y/(y-1)＞0
又：y＞0
∴y-1＞0
∴y＞1
x+y = 4y/(y-1)+y = (4y-4+4)/(y-1)+y = 4 + 4/(y-1) + y
= 5 + 4/(y-1) + (y-1)
= 5 + {2/√(y-1)-√(y-1)}² + 4
= {2/√(y-1)-√(y-1)}² + 9 ≥9
最小值9

Answer 4