Question

A function \( f : \mathbb{R} \to \mathbb{R} \), satisfies \[ \frac{f(x+y)}{3} = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all} \, x, y \in \mathbb{R}. \] If the function \( f \) is differentiable at \( x = 0 \), then \( f \) is:

• A. linear
• B. quadratic
• C. cubic
• D. biquadratic

Hint:

For functional equations of the form \( f(x+y) = f(x) + f(y) \), the solution is always a linear function \( f(x) = cx \) where \( c \) is constant, provided the function is differentiable.