Answer is: option3

Solution:

To solve this, we use the formula for the derivative of the inverse function: \[ g'(x) = \frac{1}{f'(g(x))} \] We are asked to find \( g'(25) \). First, we need to find \( g(25) \). Since \( g \) is the inverse of \( f \), we know that \( f(g(25)) = 25 \). Start by solving for \( x \) such that: \[ f(x) = 25 \] Given that \( f(x) = 3x^3 + 1 \), we solve: \[ 3x^3 + 1 = 25 \implies 3x^3 = 24 \implies x^3 = 8 \implies x = 2 \] Thus, \( g(25) = 2 \). Now, we need to compute \( f'(x) \), the derivative of \( f(x) \). Since \( f(x) = 3x^3 + 1 \), we have: \[ f'(x) = 9x^2 \] So, \[ f'(2) = 9 \times (2)^2 = 9 \times 4 = 36 \] Finally, applying the formula for the inverse function's derivative: \[ g'(25) = \frac{1}{f'(g(25))} = \frac{1}{f'(2)} = \frac{1}{36} \]

The correct answer is C) \( \frac{1}{36} \).