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. `(2+a)^(2)f''(1)+(2-a)^(2)f''(-1)=0`
B. `(2-a)^(2)f''(1)-(2+a)^(2)f''(-1)=0`
C. `f'(1)f'(-1)=(2-a)^(2)`
D. `f'(1)f'(-1)=-(2+a)^(2)`

Answer

A

Explanation

We have,
`f(x)=(x^(2)-ax+1)/(x^(2)+ax+1)`
`impliesf'(x)=((x^(2)+ax+1)(2x-a)-(x^(2)-ax+1)(2x+a))/((x^(2)+ax+1)^(2))`
`implies" "f'(x)=(2a(x^(2)-1))/((x^(2)+ax+1)^(2))`
`implies" "f''(x)=(4ax(x^(2)+ax+1)^(2)-4a(x^(2)-1)(2x+a)(x^(2)+ax+1))/((x^(2)+ax+1)^(4))`
`implies" "f''(x)=(4a{x(x^(2)+ax+1)-(x^(2)-1)(2x+a)})/((x^(2)+ax+1)^(3))`
`:." "f'(1)=0" and "f''(1)=(4a)/((2+a)^(2)),f''(-1)=-(4a)/((2-a)^(2))`
`implies" "(2+a)^(2)f''(1)+(2-a)^(2)f''(-1)=0`