Let f: R- 0 R be a function satisfying the relation f(x) + 2f ( 1 x ) = 3x + x for all x 0 , where is a real constant. If _ x x f(x) = , where is a finite real number, then the value of 2( + ) is equal to

Step-by-step solution

The given functional equation is: f(x) + 2f ( 1 x ) = 3x + x ... (1) Replace x by 1 x in equation (1): f ( 1 x ) + 2f(x) = 3 x + x ... (2) Multiply equation (2) by 2 : 2f ( 1 x ) + 4f(x) = 6 x + 2 x ... (3) Subtract equation (1) from equation (3) to eliminate f ( 1 x ) : 3f(x) = ( 6 x + 2 x ) - (3x + x ) 3f(x) = (2 - 3)x + 6 - x f(x) = 2 - 3 3 x + 6 - 3x We are given that _ x x f(x) = , where is finite. Multiply f(x) by x : x f(x) = 2 - 3 3 x^2 + 6 - 3 For the limit as x to be a finite real number, the coefficient