Home
Class 12
MATHS
If f(x) is a cubic polynomial x^3 + ax^2...

If `f(x)` is a cubic polynomial `x^3 + ax^2+ bx + c` such that `f(x)=0` has three distinct integral roots and `f(g(x)) = 0` does not have real roots, where `g(x) = x^2 + 2x - 5,` the minimum value of `a + b + c` is

A

504

B

532

C

719

D

764

Text Solution

Verified by Experts

The correct Answer is:
C
Let `alpha_(1), alpha_(2)` and `alpha_(3)` be the roots of `f(x)=0` such that and `g(x)` can take all values `[-6,oo)`
`:'g(x)=(x+1)^(2)-6ge-6`
`:.alpha_(3)le-7,alpha_(2),le-8,alpha_(1)le-9`
`:.a+b+cge719`
`:.` Minimum value of `a+b+c` is 719.
`:'alpha_(1)+alpha_(2)+alpha_(3)=-a`
`implies-ale-24`
`impliesage24`
`alpha_(1) alpha_(2)+alpha_(2)alpha_(3)+alpha_(3)alpha_(1)=b`
`impliesbge191`
and `alpha_(1)alpha_(2)alpha_(3)=-c`
`implies-cle-504`
`impliescge504`
`:.a+b+cge719`
Hence minimum value of `a+b+c` is 719.
Promotional Banner

Similar Questions

Explore conceptually related problems

If 0 and 1 are the zeroes of the polynomial f (x) = 2x ^(3) - 3x ^(2) + ax +b, then find the values of a and b.

The real number k for which the equation : 2x^(3)+3x+k=0 has two distinct real roots in [0, 1] :

If x^2+3x+5=0 a n d a x^2+b x+c=0 have common root/roots and a ,b ,c in N , then find the minimum value of a+b+ c dot

If x^(2) + ax + b = 0 and x^(2) + bx + a = 0 have a common root, then the numberical value of a + b is :

If f(x)=(a x^2+b)^3, then find the function g such that f(g(x))=g(f(x))dot

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 int xe^(2x) dx = e^(2x).f(x)+c , where c is the constant of integration, then f(x) is