Read the follwing passage and answer the questions.
Consider the polyomial `f(x)=1+2x+3x^(2)+4x^(3).` Let s be the sum of all distinct real roots of `f(x) and let t =|s|.`
The function `f'(x)` is

Options

A. `(-(1)/(4),0)`
B. `(-11,-(3)/(4))`
C. `(-(3)/(4),-(1)/(2))`
D. `(0,(1)/(4))`

Answer

C

Explanation

Given, `f(x)4x^(3)+3x^(2)+2x+1`
`f(x)=2(6x^(2)+3x+1)`
`impliesD=9-24lt0`
Hence, `f(x)=0` has only one real root.
`f(-(1)/(2))=1-1+3/4-4/8gt0`
`f(-(3)/(4))=1-6/4+27/16-108/64`
`=(64-96+108-108)/(64)lt0`
f(x) changes its sign in `(-(3)/(4),-(1)/(2))`
Hence, f(x)=0 has a root in `(-(3)/(4),-(1)/(2)).`