Consider the function `f:(-oo, oo) -> (-oo ,oo)` defined by `f(x) =(x^2 - ax + 1)/(x^2+ax+1) ;0 lt a lt 2`. Which of the following is true ?

Options

A. g'(x) is postitive on `(-oo,0)` and negative on `(0,oo)`
B. f'(x) is negative on `(-oo, 0)` and postive on `(0,oo)`
C. g'(x) chages sing on both `(-oo,0) and (0,oo)`
D. g'(x) does not change not change sign on `(-oo,oo)`

Answer

B

Explanation

we have
`f(x)=(x^2-ax+1)/(x^2+ax+1),0 gt a lt 2 `
`rArr f(x)=(2a(x^2+1))/(x^2+ax+1)^2`

The singns of f'(x) are as shown below :
`therefore g(x) gt 0 "for all "x in (1,oo) and f'(x) lt 0 " for all " x in (0,1)`
Now ,
`f(x)=underset(0)overset(e^x)int(f'(t))/(1+t^2)dt`
`rArr g(x) = e^x(f'(t))/(1+e^(2x))`
`rArr` Sign cf g' (x) is same as that of `f'(e^x)`
`rArr g'(x) gt 0 " for all " a in (0,oo) "and" g(x) lt 0 ` for all
`x in (-oo,0)`