Let \( f(x) \) be a polynomial function satisfying
\[
f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right).
\]
If \( f(4) = 65 \) and \( I_1, I_2, I_3 \) are in GP, then \( f'(I_1), f'(I_2), f'(I_3) \) are in:
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For polynomial functions satisfying specific functional equations, determining the form of the polynomial is often the first step. Derivatives of polynomials can then be analyzed to determine the nature of sequences.