Paragraph: Consider the function f:(- , ) (- , ) defined by f(x)= x^2-a x+1 x^2+a x+1 ; 0 Question: Which of the following is true ?

Step-by-step solution

f(x)= & (x^2+a x+1 )-2 a x x^2+a x+1 =1- 2 a x x^2+a x+1 f^ (x) & =- [ (x^2+a x+1 ) 2 a-2 a x(2 x+a) (x^2+a x+1 )^2 ] & =- [ -2 a x^2+2 a (x^2+a x+1 )^2 ]=2 a [ (x^2-1 ) (x^2+a x+1 )^2 ] aligned aligned & and aligned f^ (x) & =2 a [ (x^2+a x+1 )^2(2 x)-2 (x^2-1 ) (x^2+a x+1 )(2 x+a) (x^2+a x+1 )^4 ] & =2 a [ 2 x (x^2+a x+1 )-2 (x^2-1 )(2 x+a) (x^2+a x+1 )^3 ] aligned & Now, f^ (1)= 4 a(a+2) (a+2)^3 = 4 a (a+2)^2 aligned and f(-1)= 4 a(a-2) (2-a)^3 =- 4 a (a-2)^2 (2+a)^2 f^ (1)+(2-a)^2 f^ (-1)=4 a-4 a=0