Limits, Continuity & Differentiability · Functional equation and differentiability

Let

f : \mathbb{R} \to \mathbb{R}

be a differentiable function such that

f\left(\frac{x + y}{3}\right) = \frac{f(x) + f(y)}{3}\quad\text{for all } x, y \in \mathbb{R}

and

f'(0) = 3

. Then the minimum value of the function

g(x) = 3 + e^x f(x)

, is

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