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Let x,y,z in R^(+) and 2xy+3yz+4xz=18. I...

Let `x,y,z in R^(+)` and `2xy+3yz+4xz=18`. If `alpha`, `beta` and `gamma` be the values of `x`, `y` and `z` respectively, for which `xyz` attains its maximum value, then the value of `2alpha+beta+gamma=`

A

`4`

B

`6`

C

`8`

D

`12`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` Using `A.M. ge G.M.`
`(2xy+3yz+4xz)/(3) ge (24x^(2)y^(2)z^(2))^((1)/(3))`
`implies216 ge 24x^(2)y^(2)z^(2)`
`impliesx^(2)y^(2)z^(2) le 9`
`impliesxyz le 3`
So, `xyz` is greatest when `2xy=3yz=4xz=6`
`impliesxy=3`, `yz=2`, `zx=3//2`
`impliesalpha=3//2`, `beta=2`, `gamma=1`
`2alpha+beta+gamma=6`
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