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If 2x^(3)+ax^(2)+bx+4=0 (a and b are pos...

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

Text Solution

Verified by Experts

The correct Answer is:
C
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
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