If `2x^(3)+ax^(2)+bx+4=0` (a and b are positive real numbers) has three real roots.
The minimum value of `b^(3)` is

Options

A. 432
B. 864
C. 1728
D. none of these

Answer

B

Explanation

Let `alpha, beta` and `gamma` be the roots of `2x^(3)+ax^(2)+bx+4=0`
`:.alpha +beta+gamma=-a/2`
`alpha beta+beta gamma +gamma alpha =b/2` and `alpha beta gamma =-2`
`:'AMgeGM`
`:.((-alpha)(-beta)+(-beta)(-gamma)+(-gamma)(-alpha))/3`
`ge{(-alpha)(-beta)(-gamma)(-gamma)(-alpha)}^(1//3)`
`=(b//2)/3ge(4)^(1//3)`
`impliesbge6(4)^(1//3)`.....ii
or `b^(3)ge864`
Hence minimum value of `b^(3)` is 964