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Consider the polynomial f(x)=1+2x+3x^2+4...

Consider the polynomial `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|.

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)

Text Solution

Verified by Experts

The correct Answer is:
B
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)`
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