If `x,y,z` be three positive numbers such that `xyz^(2)` has the greatest value `(1)/(64)`, then the value of `(1)/(x)+(1)/(y)+(1)/(z)` is

Options

A. `a=b=(1)/(2),c=(1)/(4)`
B. `a=b=(1)/(4),c=(1)/(2)`
C. `a=b=c=(1)/(3)`
D. none of these

Answer

B

Explanation

Using `A.M.geG.M.,` we have
`(a+b+(c)/(2)+(c)/(2))/4ge(a.b.(c)/(2).(c)/(2))^(1//4)`
`implies" "(a+b+c)/(4)ge((abc^(2))/(4))^(1//4)`
`implies" "(a+b+c)^(4)ge64abc^(2)`
`implies" "abc^(2)le(1)/(64)(a+b+c)^(4)`
But, `abc^(2)` has the greatest value equal to `(1)/(64)`.
`:." "(1)/(64)(a+b+c)^(4)=(1)/(64)impliesa+b+c=1`
Also, `abc^(2)=4(a.b.(c)/(2).(c)/(2))` is greatest when
`a=b=(c)/(2)=(c)/(2)impliesa=b=(c)/(2)`
From (i) and (ii), we get`a=b=(1)/(4),c=(1)/(2).`