Question

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

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

Hint:

Mean-type functional equations: Symmetric averaging usually implies linearity. Try polynomial substitution and compare degrees.

Answer 1

The Correct Option is A