If `2x^(3)+ax^(2)+bx+4=0` (a and b are positive real numbers) has three real roots.
The minimum value of `a^(3)` is a. 108 b. 216 c. 432 d. 864

Options

A. 108
B. 216
C. 432
D. 864

Answer

C

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)+(-gamma))/3ge{(-alpha)(-beta)(-gamma)}^(1//3)`
`implies(a/2)/3ge(2)^(1//3)`
`:.age6(2)^(1//3)`....i
or `a^(3)ge432` ltbRgt Hence minimum value of `a^(3)` is 432