If `alpha` is the only real root of the equation `x^3 + bx^2 + cx + 1 = 0 (b < c)`, then the value of `tan^-1 alpha+ tan^-1 (alpha^-1)` is equal to :

Options

A. `(pi)/(2)`
B. `-(pi)/(2)`
C. 0
D. non existent

Explanation

Let `f(x)=x^(3)+bx^(2)+cx+1` Then
`f(0)=1 gt 0 and f(-1)=b-c lt 0`
`rarr alpha lt 0`
`rarr tan^(-1)((1)/(alpha))=-pi + cot^(-1)alpha`
`rarr tan^(-1) alpha+tan^(-1)(1)/(alpha)=-pi + tan^(-1)alpha+cos^(-1)alpha`

`rarr tan^(-1)alpha+tan^(-1)(1)/(alpha)=-pi++(pi)/(2)=-(pi)/(2)`