Let f: R R be a function satisfying f(x) + 2f(1-x) = 3x^2 for all x R . The area of the region bounded by the curve y = f(x) and the line y = -1 is :

Step-by-step solution

Given functional equation: f(x) + 2f(1-x) = 3x^2 (1) Replace x with 1-x : f(1-x) + 2f(x) = 3(1-x)^2 (2) To eliminate f(1-x) , multiply equation (2) by 2 and subtract equation (1): 2[f(1-x) + 2f(x)] - [f(x) + 2f(1-x)] = 6(1-x)^2 - 3x^2 4f(x) - f(x) = 6(1 - 2x + x^2) - 3x^2 3f(x) = 3x^2 - 12x + 6 f(x) = x^2 - 4x + 2 To find the points of intersection of the curve y = f(x) and the line y = -1 , we set: x^2 - 4x + 2 = -1 x^2 - 4x + 3 = 0 (x - 1)(x - 3) = 0 x = 1, 3 The area bounded by the curve and the line is given by