Let `f(x) = (1-x)^(2) sin^(2)x+ x^(2)` for all `x in IR` and let `g(x) = underset(1)overset(x)int((2(t-1))/(t+1)-lnt) f(t)` dt for all `x in (1,oo)`.
Consider the statements :
P : There exists some `x in IR` such that `f(x) + 2x = 2 (1+x^(2))`
Q : There exist some `x in IR` such that `2f(x) + 1 = 2x(1+x)`
Then

Answer

B

Explanation

We have `f(x)=(1 -x)^2 sin ^2 x+x^2`
and `g(x) =underset(1)overset(x){(2(t-1))/(x+1)-log_e t}f(t) dt `
`therefore g'(x)={(2(x-1))/(x+1)-log_ex }f(x)`
Let `h(x)={(2(x-1))/(x+1)-log_ex}={2-(4)/(x+1)-log_ex}` Then
`f(x)={(4)/(x+1)^2-1/x}= -((x-1)^2)/((x +1)^2x)lt 0 "for all"x gt 1 `
So h(x) is decreasing for all ` x gt 1`
`therefore h(x) lt h (1) " for all " x gt 1`
`rArr h(x) lt 0 "for all " x gt 1`
`rArr h(x) f(x) lt 0 " for all " " "[ because f(x)=(1-x^2) sin^2x+x^2 gt 0 for all x in (1,oo)]`
`rArr g'(x) lt 0 " for all " x gt 1 " "[ because g(x)= h(x) f(x)]`
`rArr g ` is decreasing on `(1,oo)`