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. The formula we will use is:

\[ g'(y) = \frac{1}{f'(g(y))} \]

### Step 1: Find \( x \) such that \( f(x) = 5 \)

We need to solve the equation:

\[ x^3 + 3x + 1 = 5 \]

This simplifies to:

\[ x^3 + 3x - 4 = 0 \]

### Step 2: Solve the cubic equation

To find the roots of the equation \( x^3 + 3x - 4 = 0 \), we can try substituting some values. Testing \( x = 1 \):

\[ 1^3 + 3(1) - 4 = 1 + 3 - 4 = 0 \]

So, \( x = 1 \) is a root. Therefore, \( f(1) = 5 \).

### Step 3: Find \( f'(x) \)

Next, we need to find the derivative \( f'(x) \):

\[ f'(x) = 3x^2 + 3 \]

### Step 4: Evaluate \( f'(1) \)

Now, we will evaluate \( f'(1) \):

\[ f'(1) = 3(1)^2 + 3 = 3 + 3 = 6 \]

### Step 5: Use the inverse function derivative formula

Now we can find \( g'(5) \):

\[ g'(5) = \frac{1}{f'(1)} = \frac{1}{6} \]

### Final Answer

Thus, the value of \( g'(5) \) is:

\[ \boxed{\frac{1}{6}} \]