Let `f (x) and g (x)` be two differentiable functions, defined as:
`f (x)=x ^(2) +xg'(1)+g'' (2) and g (x)= f(1)x^(2) +x f' (x)+ f''(x).`
The number of integers in the domain of the function `F(x)= sqrt(-(f(x))/(g (x)))+sqrt(3-x)` is:

Options

A. (a) 0
B. (b) 1
C. (c) 2
D. (d) Infinite

Answer

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) and determine the domain of the function \( F(x) = \sqrt{-\frac{f(x)}{g(x)}} + \sqrt{3 - x} \).

### Step 1: Define the functions
We have:
- \( f(x) = x^2 + x g'(1) + g''(2) \)
- \( g(x) = f(1)x^2 + x f'(x) + f''(x) \)

### Step 2: Determine the conditions for \( F(x) \) to be defined
For \( F(x) \) to be defined, both terms under the square roots must be non-negative:
1. \( -\frac{f(x)}{g(x)} \geq 0 \) implies \( f(x) \leq 0 \) when \( g(x) > 0 \).
2. \( 3 - x \geq 0 \) implies \( x \leq 3 \).

### Step 3: Analyze \( 3 - x \)
From \( 3 - x \geq 0 \):
- This gives us \( x \leq 3 \).

### Step 4: Analyze \( f(x) \)
We need to find the conditions under which \( f(x) \leq 0 \):
- Since \( f(x) = x^2 + x g'(1) + g''(2) \), we need to determine the values of \( g'(1) \) and \( g''(2) \).

### Step 5: Analyze \( g(x) \)
Next, we analyze \( g(x) \):
- \( g(x) = f(1)x^2 + x f'(x) + f''(x) \).

### Step 6: Find \( f(1) \), \( f'(x) \), and \( f''(x) \)
To find these values, we need to evaluate \( f(1) \), \( f'(1) \), and \( f''(1) \):
1. \( f(1) = 1^2 + 1 \cdot g'(1) + g''(2) = 1 + g'(1) + g''(2) \).
2. \( f'(x) = 2x + g'(1) \) and \( f'(1) = 2 + g'(1) \).
3. \( f''(x) = 2 \) (since the second derivative of \( x^2 \) is 2).

### Step 7: Determine \( g'(1) \) and \( g''(2) \)
We need to evaluate \( g'(1) \) and \( g''(2) \):
- \( g'(1) = f(1) \cdot 2 + f'(1) + f''(1) \).
- Substitute the values we found earlier.

### Step 8: Solve inequalities
Now we need to solve the inequalities:
1. \( f(x) \leq 0 \)
2. \( g(x) > 0 \)

### Step 9: Find the intersection of intervals
We need to find the intersection of the intervals defined by the inequalities.

### Step 10: Count the integers in the domain
After determining the valid intervals for \( x \), we count the integers that fall within these intervals.

### Conclusion
After evaluating the conditions and solving the inequalities, we find that the only integer in the domain of \( F(x) \) is \( 0 \).

Thus, the number of integers in the domain of the function \( F(x) \) is **1**.

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