Let `a, b, c in R` with `a gt 0` such that the equation `ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0` has non-real roots.
If `P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b`, then

Options

A. `P(x) gt 0` for all `x in R` and `Q(x) lt 0` for all `x in R`.
B. `P(x) lt 0` for all `x in R and Q(x) gt 0` for all `x in R`.
C. neither `P(x) gt 0` for all `x in R` nor `Q(x) gt 0` for all `x in R`.
D. exactly one of P(x) or Q(x) is positive for all real x.

Answer

4

Explanation

Given equation has non - real roots.
`therefore D lt 0`
`rArr b^(2)C^(2) - 4(b^(3) + c^(3) - 4abc) a lt 0`
`rArr (b^(2)c^(2) - 4ab^(3)) + (16a^(2)bc - 4ac^(3)) lt 0`
`rArr b^(2)(c^(2) - 4ab) - 4ac(c^(2) - 4ab) lt 0`
`rArr (b^(2) - 4ac) (c^(2) - 4ab) lt 0`
`rArr D_(P(x)) * D_(Q(x)) lt 0`
Therefore, exactly one of P(x) or Q(x) is positive for all real x.