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If x,y in R^(+) such that x + y = 8, the...

If `x,y in R^(+)` such that x + y = 8, then find the minimum value of `(1 + (1)/(x)) (1 + (1)/(y))`

Text Solution

Verified by Experts

The correct Answer is:
`(25)/(16)`
`(+(1)/(x))(1+(1)/(y))=(1+x+y+xy)/(xy)=(9+xy)/(xy)=(9)/(xy)+1`.
Now,using `A.M ge G.M`., we have
`(x+y)/(2)ge sqrt(xy)`
`rArr 4 ge sqrt( xy)`
`rArr 16ge xy`
`rArr (1)/(xy) ge (1)/(16)`
`rArr (9)/(xy)+1 ge (25)/(16)`
Therefore, the minimum value of `(1+(1)/(x))(1+(1)/(y))` is `(25)/(16)`.
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