Step 1: Determine the form of \( f(x) \).
The given functional equation is:
\[
f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right).
\]
Rearranging yields:
\[
(f(x) - 1)(f\left(\frac{1}{x}\right) - 1) = 1.
\]
This implies \( f(x) - 1 \) is of the form \( x^n \), where \( n \) is an integer.
Let:
\[
f(x) = x^n + 1.
\]
Step 2: Use the given condition \( f(4) = 65 \).
\[
f(4) = 4^n + 1 = 65 \quad \Rightarrow \quad 4^n = 64 \quad \Rightarrow \quad n = 3.
\]
Therefore, \( f(x) = x^3 + 1 \).
Step 3: Compute the derivative \( f'(x) \).
\[
f'(x) = 3x^2.
\]
Step 4: Analyze \( f'(I_1), f'(I_2), f'(I_3) \).
Given \( I_1, I_2, I_3 \) are in GP, let:
\[
I_2 = I_1 r, \quad I_3 = I_1 r^2,
\]
where \( r \) is the common ratio.
Then:
\[
f'(I_1) = 3I_1^2, \quad f'(I_2) = 3I_2^2 = 3I_1^2 r^2, \quad f'(I_3) = 3I_3^2 = 3I_1^2 r^4.
\]
The sequence \( f'(I_1), f'(I_2), f'(I_3) \) is:
\[
3I_1^2, 3I_1^2 r^2, 3I_1^2 r^4,
\]
which constitutes a geometric progression (GP).
Step 5: Determine the correct option.
The sequence \( f'(I_1), f'(I_2), f'(I_3) \) is in GP.