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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+b)^(3)` is

A

1728

B

3456

C

6912

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`
From Eqs (i) and (ii) we get
`abge6(2)^(1//3).6(4)^(1//3)`
`impliesabge36xx2`
`:'(a+b)/2gesqrt(ab)ge6sqrt(2)implies(a+b)/2ge6sqrt(2)`
`:.a+bge12sqrt(2)`
or `(a+b)^(3)ge3456sqrt(2)`
Hence minimum value of `(a+b)^(3)` is `3456sqrt(2)`
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