Suppose that `f(x)` is differentiable invertible function `f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot` Given that `f(1)=f^(prime)(1)=1,h(1)=0` and `g(x)` is inverse of `f(x)` . Let `G(x)=x^2g(x)-x h(g(x))AAx in Rdot` Which of the following is/are correct? `G^(prime)(1)=2` b. `G^(prime)(1)=3` c.`G^(1)=2` d. `G^(1)=3`

Options

A. G''(1) = 2
B. G'(1) = 3
C. G''(1) = 2
D. G''(1) = 3

Explanation

To solve the problem step by step, we will analyze the given information and derive the necessary equations to find \( G'(1) \).

### Step 1: Understanding the Functions
We are given:
- \( f(x) \) is a differentiable and invertible function with \( f'(x) \neq 0 \).
- \( h'(x) = f(x) \).
- \( f(1) = 1 \), \( f'(1) = 1 \), and \( h(1) = 0 \).
- \( g(x) \) is the inverse of \( f(x) \).

### Step 2: Finding \( g(1) \)
Since \( g(x) \) is the inverse of \( f(x) \), we have:
\[ f(g(x)) = x \]
Substituting \( x = 1 \):
\[ f(g(1)) = 1 \]
Given \( f(1) = 1 \), it follows that:
\[ g(1) = 1 \]

### Step 3: Finding \( g'(1) \)
Differentiating both sides of \( f(g(x)) = x \) with respect to \( x \):
\[ f'(g(x)) \cdot g'(x) = 1 \]
Substituting \( x = 1 \):
\[ f'(g(1)) \cdot g'(1) = 1 \]
Since \( g(1) = 1 \) and \( f'(1) = 1 \):
\[ 1 \cdot g'(1) = 1 \implies g'(1) = 1 \]

### Step 4: Defining \( G(x) \)
Now we define:
\[ G(x) = x^2 g(x) - x h(g(x)) \]

### Step 5: Finding \( G'(x) \)
To differentiate \( G(x) \), we apply the product rule and chain rule:
\[ G'(x) = \frac{d}{dx}(x^2 g(x)) - \frac{d}{dx}(x h(g(x))) \]
Using the product rule on both terms:
\[ G'(x) = 2x g(x) + x^2 g'(x) - (h(g(x)) + x h'(g(x)) g'(x)) \]

### Step 6: Evaluating \( G'(1) \)
Substituting \( x = 1 \):
\[ G'(1) = 2 \cdot 1 \cdot g(1) + 1^2 g'(1) - (h(g(1)) + 1 \cdot h'(g(1)) g'(1)) \]
We know:
- \( g(1) = 1 \)
- \( g'(1) = 1 \)
- \( h(g(1)) = h(1) = 0 \)
- \( h'(g(1)) = h'(1) = f(1) = 1 \)

Now substituting these values:
\[ G'(1) = 2 \cdot 1 + 1 - (0 + 1 \cdot 1 \cdot 1) \]
This simplifies to:
\[ G'(1) = 2 + 1 - 1 = 2 \]

### Conclusion
Thus, we find that:
\[ G'(1) = 2 \]

### Final Answer
The correct option is:
- **a. \( G'(1) = 2 \)**