Question

If \( f(x) \) satisfies the relation \( f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3} \) and \( f(0) = 3 \), then the minimum value of \( g(x) = 3 + e^x f(x) \) is:

• A. \( \frac{3(e-1)}{e} \)
• B. \( \frac{(e-1)}{e} \)
• C. \( \frac{(e-1)}{3} \)
• D. \( \frac{e(e-1)}{3} \)

Hint:

Functional equations often simplify by substituting \( y=0 \) or \( y=x \). If the expression involves \( e^x f(x) \), check for points where the derivative vanishes to locate the minimum.

Answer 1

The Correct Option is A