Let f : R R be a differentiable function satisfying f(x+y) = f(x) + f(y) + 3x^2y + 3xy^2 for all x, y R . If _ x 0 f(x) x = 1 and g(x) is the inverse function of f(x) , then the value of |2197 g''(10)| is equal to _____.

Step-by-step solution

Given _ x 0 f(x) x = 1 . Since the limit exists and the denominator approaches 0 , the numerator must also approach 0 , so f(0) = 0 . Applying L'Hopital's rule, _ x 0 f'(x) = f'(0) = 1 . Differentiating the given functional equation f(x+y) = f(x) + f(y) + 3x^2y + 3xy^2 partially with respect to x (treating y as a constant): f'(x+y) = f'(x) + 6xy + 3y^2 Substituting x = 0 : f'(y) = f'(0) + 0 + 3y^2 = 1 + 3y^2 Thus, f'(x) = 3x^2 + 1 . Integrating with respect to x : f(x) = x^3 + x + C Since f(0) = 0 , C = 0 , so f(x)