Let ` f (x) = x^(3)+ 4x ^(2)+ 6x and g (x)` be inverse then the vlaue of `g' (-4):`

Options

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

Answer

To find the value of \( g'(-4) \) where \( g(x) \) is the inverse of the function \( f(x) = x^3 + 4x^2 + 6x \), we can follow these steps:

### Step 1: Understand the relationship between \( f \) and \( g \)
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ g(f(x)) = x \]
Differentiating both sides with respect to \( x \) gives:
\[ g'(f(x)) \cdot f'(x) = 1 \]
From this, we can express \( g' \) in terms of \( f' \):
\[ g'(f(x)) = \frac{1}{f'(x)} \]

### Step 2: Find \( x \) such that \( f(x) = -4 \)
We need to find the value of \( x \) for which \( f(x) = -4 \):
\[ f(x) = x^3 + 4x^2 + 6x = -4 \]
Rearranging gives:
\[ x^3 + 4x^2 + 6x + 4 = 0 \]

### Step 3: Solve the cubic equation
We can attempt to factor or find roots of the equation:
\[ x^3 + 4x^2 + 6x + 4 = 0 \]
By inspection or using the Rational Root Theorem, we can check for simple roots. Testing \( x = -2 \):
\[ (-2)^3 + 4(-2)^2 + 6(-2) + 4 = -8 + 16 - 12 + 4 = 0 \]
Thus, \( x = -2 \) is a root.

### Step 4: Factor the cubic polynomial
Now we can factor the cubic polynomial:
\[ x^3 + 4x^2 + 6x + 4 = (x + 2)(x^2 + 2x + 2) \]
The quadratic \( x^2 + 2x + 2 \) has no real roots (discriminant is negative), so the only real solution is \( x = -2 \).

### Step 5: Find \( f'(-2) \)
Next, we need to compute \( f'(-2) \):
\[ f'(x) = 3x^2 + 8x + 6 \]
Now substituting \( x = -2 \):
\[ f'(-2) = 3(-2)^2 + 8(-2) + 6 = 3(4) - 16 + 6 = 12 - 16 + 6 = 2 \]

### Step 6: Calculate \( g'(-4) \)
Now we can find \( g'(-4) \):
\[ g'(-4) = \frac{1}{f'(-2)} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( g'(-4) \) is:
\[ \boxed{\frac{1}{2}} \]