在括号里填上和为1的两个正数，使1()+9()的值最小，则这两个正数的积等于．

Question

(1)/(())+(9)/(()) 的值最小，则这两个正数的积等于______．(1)/(()11((+(9)/(()) 的值最小，则这两个正数的积等于______．(9)/(()99((

Answer 1

设所添的两个数分别为x，y，则x>0，y>0且x+y=1
(1)/(x)+(9)/(y)=（(1)/(x)+(9)/(y)）（x+y）=10+(y)/(x)+(9x)/(y)≥10+2√((y)/(x)•(9x)/(y))=16
当且仅当(y)/(x)＝(9x)/(y)且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (1)/(x)111xxx+(9)/(y)999yyy=（(1)/(x)+(9)/(y)）（x+y）=10+(y)/(x)+(9x)/(y)≥10+2√((y)/(x)•(9x)/(y))=16
当且仅当(y)/(x)＝(9x)/(y)且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (1)/(x)111xxx+(9)/(y)999yyy）（x+y）=10+(y)/(x)+(9x)/(y)≥10+2√((y)/(x)•(9x)/(y))=16
当且仅当(y)/(x)＝(9x)/(y)且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (y)/(x)yyyxxx+(9x)/(y)9x9x9xyyy≥10+2√((y)/(x)•(9x)/(y))=16
当且仅当(y)/(x)＝(9x)/(y)且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) ≥10+2√((y)/(x)•(9x)/(y))

(y)/(x)•(9x)/(y)

(y)/(x)•(9x)/(y)(y)/(x)yyyxxx•(9x)/(y)9x9x9xyyy=16
当且仅当(y)/(x)＝(9x)/(y)且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (y)/(x)yyyxxx＝(9x)/(y)9x9x9xyyy且x+y=1即x=(1)/(4)，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (1)/(4)111444，y=(3)/(4)时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) (3)/(4)333444时取等号
∴xy＝(1)/(4)•(3)/(4)=(3)/(16)
故答案为：(3)/(16) xy＝(1)/(4)111444•(3)/(4)333444=(3)/(16)
故答案为：(3)/(16) (3)/(16)333161616
故答案为：(3)/(16) (3)/(16)333161616