If `f(1)=3, f'(1) = 2, f''(1)=4`, then `(f^-)''(3)=` (where `f^-1=`inverse of `y = f(x))`

Options

A. 1
B. `-(1)/(2)`
C. `-2`
D. None of these

Explanation

To solve the problem, we need to find the second derivative of the inverse function \( (f^{-1})''(3) \) given the values of \( f(1) = 3 \), \( f'(1) = 2 \), and \( f''(1) = 4 \).

### Step-by-Step Solution:

1. **Identify the relationship between the derivatives of \( f \) and \( f^{-1} \)**:
The derivatives of the inverse function can be expressed using the following formulas:
- The first derivative of the inverse function is given by:
\[ (f^{-1})'(y) = \frac{1}{f'(x)} \quad \text{where } y = f(x) \]
- The second derivative of the inverse function is given by:
\[ (f^{-1})''(y) = -\frac{f''(x)}{(f'(x))^3} \]

2. **Determine \( x \) such that \( f(x) = 3 \)**:
From the given information, we know that \( f(1) = 3 \). Therefore, \( x = 1 \) when \( y = 3 \).

3. **Substitute the known values into the formulas**:
- We have \( f'(1) = 2 \) and \( f''(1) = 4 \).
- Now, we can substitute these values into the formula for the second derivative of the inverse function:
\[ (f^{-1})''(3) = -\frac{f''(1)}{(f'(1))^3} \]
- Substituting the values:
\[ (f^{-1})''(3) = -\frac{4}{(2)^3} \]

4. **Calculate the result**:
- Calculate \( (2)^3 = 8 \).
- Therefore:
\[ (f^{-1})''(3) = -\frac{4}{8} = -\frac{1}{2} \]

### Final Answer:
\[ (f^{-1})''(3) = -\frac{1}{2} \]