Don't Even Try Guessing

Algebra Level 2

Let f(x)=a0+a1x+........+an1xn1+anxnf(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)=3f(9)=3 and f(81)=27f(81)=27. Find the maximum possible achievable value of f(27)f(27).

Extra Credit: Find all polynomials having the equality case.

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