Question

已知正数x y满足x+2y=2 则x+8y/xy 的最小值为

Answer 1

∵正数x y满足x+2y=2,

∴2=x+2y=x/2+x/2+2y≥3[³√(x²y/2)]

即³√(x²y/2)≤2/3,

x²y/2≤8/27,

x²y≤16/27,

1/(x²y)≥27/16,

∴(x+8y)/(xy)=1/y+8/x=1/y+4/x+4/x≥3[³√(16/x²y)≥3(³√27)=9

当且仅当x/2=2y=2/3,即x=4/3,y=1/3时,(x+8y)/(xy)取得最小值9.