Let f: R R be a function satisfying f(x) + 3f(-x) = e^x + (x) for all x R . If _ x 0 8f(x) - + x x^2 = , where , , R and is finite, then the value of + + is equal to

Step-by-step solution

The given functional equation is: f(x) + 3f(-x) = e^x + (x) ... (1) Replace x by -x in equation (1): f(-x) + 3f(x) = e^ -x + (-x) = e^ -x - (x) ... (2) Multiply equation (2) by 3 : 3f(-x) + 9f(x) = 3e^ -x - 3 (x) ... (3) Subtract equation (1) from equation (3) to eliminate f(-x) : 8f(x) = 3e^ -x - 3 (x) - e^x - (x) 8f(x) = 3e^ -x - e^x - 4 (x) To evaluate the limit as x 0 , we use the Maclaurin series expansions of e^ -x , e^x , and (x) up to the x^2 term: e^ -x = 1 - x + x^2 2 - e^x = 1 + x + x^2 2 + (x) = x - x^3