If `f((x+y)/3)=(2+f(x)+f(y))/3` for all real `xa n dy` and `f^(prime)(2)=2,` then determine `y=f(x)dot`

Explanation

To solve the problem step by step, we need to analyze the given functional equation and the derivative condition.

### Step 1: Understanding the Functional Equation
We start with the functional equation:
\[ f\left(\frac{x+y}{3}\right) = \frac{2 + f(x) + f(y)}{3} \]
This holds for all real \(x\) and \(y\).
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