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. `6`
B. `8`
C. `10`
D. `12`

Answer

C

Explanation

`(c )` Consider the four positive numbers `x`, `y`,`(z)/(2)`, `(z)/(2)`
Using `A.M ge G.M.`, we have
`(x+y+(z)/(2)+(z)/(2))/(4) + ge ((xyz^(2))/(4))^((1)/(4))`
`impliesxyz^(2) le ((x+y+z)^(4))/(64)`
But given that greatest value of `xyz^(2)` is `(1)/(64)`
`impliesx+y+z=1`
But greatest value is attained when `x=y=(z)/(2)`
`impliesx=(1)/(4)`, `y=(1)/(4)`, `z=(1)/(2)`