Don't Even Try Guessing

Algebra Level 2

Let \(f(x)=a_0+a_1x+........+a_{n-1}x^{n-1}+a_nx^n\) be a polynomial with non-negative real coefficients, such that \(f(9)=3\) and \(f(81)=27\). Find the maximum possible achievable value of \(f(27)\).

Extra Credit: Find all polynomials having the equality case.

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