If `f(x)=x^(3)+3x+1 and g(x)` is the inverse function of f(x), then the value of g'(5) is equal to

Options

A. 3
B. `(1)/(3)`
C. (1)/(6)`
D. 6

Answer

To find the value of \( g'(5) \) where \( g(x) \) is the inverse function of \( f(x) = x^3 + 3x + 1 \), we can use the relationship between the derivatives of inverse functions.

### Step-by-Step Solution:

1. **Understand the relationship between \( f \) and \( g \)**:
Since \( g \) is the inverse of \( f \), we have:
\[ g(f(x)) = x \]
Differentiating both sides with respect to \( x \) gives:
\[ g'(f(x)) \cdot f'(x) = 1 \]

2. **Find \( f'(x) \)**:
We need to compute the derivative of \( f(x) \):
\[ f(x) = x^3 + 3x + 1 \]
Using the power rule, we find:
\[ f'(x) = 3x^2 + 3 \]

3. **Determine \( x \) such that \( f(x) = 5 \)**:
We need to find the value of \( x \) for which \( f(x) = 5 \):
\[ x^3 + 3x + 1 = 5 \]
Simplifying this gives:
\[ x^3 + 3x - 4 = 0 \]
Testing \( x = 1 \):
\[ 1^3 + 3(1) - 4 = 1 + 3 - 4 = 0 \]
Thus, \( f(1) = 5 \).

4. **Substitute \( x = 1 \) into the derivative**:
Now, we substitute \( x = 1 \) into \( f'(x) \):
\[ f'(1) = 3(1)^2 + 3 = 3 + 3 = 6 \]

5. **Use the relationship to find \( g'(5) \)**:
From the derivative relationship, we have:
\[ g'(f(x)) = \frac{1}{f'(x)} \]
Therefore:
\[ g'(5) = \frac{1}{f'(1)} = \frac{1}{6} \]

### Final Answer:
\[ g'(5) = \frac{1}{6} \]