Consider the polynomial
`f(x) = 1+2x+3x^(2)+4x^(3)`
Let s be the sum of all distinct real roots of `f(x)` end let `t = |s|`
The real number s lies in the interval.

Options

A. increasing in (-t,-1/4)and decreasing in (-1 /4 t)
B. decreasing (-t ,-1 /4) and increasing in (-1/4,t)
C. incresing in (-t ,t)
D. decreasing (-t, t)

Answer

B

Explanation

In example 57, we have seen that `s in (-3/4,-1/2)`
` therefore t in (1//2,3//4)`
Now `f(x)=1+2x+3x^2+4x^3`
`rArr f'(x)=2+6x+12x^2 and f''(x)=6+24x`
`therefore f'(x)= 0 rArr x =1/4`

The signs of f''(x) are as shown below.
Clearly f'(x) is increasing on `(-1/4 ,oo)` and decresing on `(-oo,-1//4,t)`