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

Options

A. 504
B. 532
C. 719
D. 764

Answer

C

Explanation

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.