Let f: R → R be a function satisfying f(x + y) = f(x) + f(y) + xy for all real x and y. Suppose f is differentiable. Which of the following could be f(x)?
Correct: B
Assume f(x) = ax² + bx + c. Substitute into the equation: a(x+y)² + b(x+y) + c = ax² + bx + c + ay² + by + c + x y. Expanding and comparing coefficients gives a = -1/2, b = 1. The constant c cancels out. Only option B matches this form.