If a, b and c are positive, then the minimum value of `a^(log b- logc) + b^(log c -loga ) + c^(log a - log b)` is :

Options

A. 3
B. 1
C. 9
D. 16

Answer

A

Explanation

Using `A.M.geG.M.,` we have
`(a^(logb-logc)+b^(logc-loga)+c^(loga-logb))/(3)`
`ge{a^(logb-logc)xxb^(logc-loga)xxc^(loga-logb)}^(1//3)`
Let `a^(logb-logc)xxb^(logc-loga)xxc^(loga-logb)=lambda`.Then,
`loglambda=(logb-logc)loga+(logc-loga)logb+(loga-logb)logc`
`implies" "loglambda=0implieslambda=1`
`a^(logb-logc)+b^(logc-loga)+c^(loga-logb)ge3`
Hence, the minimum value of
`a^(logb-logc)+b^(lobc-loga)+c^(loga-logb)`is 3.