`f(x)=x^(2)+xg'(1)+g''(2)and g(x)=f(1)x^(2)+xf'(x)+f'(x).`
The value of g(0) is

Options

A. 0
B. -3
C. 2
D. none of these

Explanation

To solve the problem, we need to find the value of \( g(0) \) given the functions \( f(x) \) and \( g(x) \).

### 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: Substitute values for \( g'(1) \) and \( g''(2) \)
Let:
\[ g'(1) = a \quad \text{and} \quad g''(2) = b \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = x^2 + ax + b \]

### Step 3: Calculate \( f'(x) \) and \( f''(x) \)
Now, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2 + ax + b) = 2x + a \]
\[ f''(x) = \frac{d^2}{dx^2}(x^2 + ax + b) = 2 \]

### Step 4: Calculate \( f(1) \)
Next, we find \( f(1) \):
\[ f(1) = 1^2 + a \cdot 1 + b = 1 + a + b \]

### Step 5: Substitute into \( g(x) \)
Now we substitute \( f(1) \), \( f'(x) \), and \( f''(x) \) into \( g(x) \):
\[ g(x) = (1 + a + b)x^2 + x(2x + a) + 2 \]
Expanding this gives:
\[ g(x) = (1 + a + b)x^2 + (2x^2 + ax) + 2 \]
Combining like terms:
\[ g(x) = (1 + a + b + 2)x^2 + ax + 2 \]
\[ g(x) = (3 + a + b)x^2 + ax + 2 \]

### Step 6: Find \( g(0) \)
To find \( g(0) \):
\[ g(0) = (3 + a + b) \cdot 0^2 + a \cdot 0 + 2 = 2 \]

### Conclusion
Thus, the value of \( g(0) \) is:
\[ \boxed{2} \]