Home
Class 12
MATHS
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)`
Promotional Banner

Similar Questions

Explore conceptually related problems

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

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

If a, b, c are positive real numbers, then the number of real roots of the equation a x^(2)+b|x|+c=0 is

If a,b,c and d are four positive real numbers such that abcd=1 , what is the minimum value of (1+a)(1+b)(1+c)(1+d) .

If a,b,c are positive real numbers, then the roots of the equation ax^(2) + bx + c =0

If three positive real numbers a,b,c are in A.P. such that abc=4, then the minimum value of b is

y=3x is tangent to the parabola 2y=ax^2+b . The minimum value of a+b is

If the equation x^(4)+px^(3)+qx^(2)+rx+5=0 has four positive real roots, find the maximum value of pr .

If x is real, the minimum value of ((2x^(2)+4x+5)/(2x^(2)+4x+7)) is

x and b are real numbers. If b gt 0 and |x| gt b , then