If `f(x)=(P^(2)-1)/(P^(2)+1)x^(3)-3x+log2` is a decreasing function of x in R then the set of possible values of P (independent of x ) is

Options

A. [-1,1]
B. `[1,oo]`
C. `[-oo,-1]`
D. none of these

Answer

A

Explanation

We have `f(x)=(a^2-1)/(a^2+1)x^3-3x+2log_e5 " for all " x in R`
`rArr f'(x) = 3((a^2-1)/(a^2+1))x^2-3 " for all " x in R`
For f(x) to be decreasing on R , we must have
`f'(x)lt0 " for all " x in R`
`rArr f'(x) = 3((a^2-1)/(a^2+1))x^2-3 " for all "x in R`
`((a^2-1)/(a^2+1))x^2-1 " for all "x in R`
`rArr (a^2-1)/(a^2+1)le0 rArra^2-1le0rArra in [-1,1]`