Let `f(x)=x^(2)+xg'(1)+g''(2) and g(x)=f(1).x^(2)+xf'(x)+f''(x)` then

Options

A. `f'(1)+f'(2)=0`
B. `g'(2)=g'(1)`
C. `g''(2)+f''(3)=6`
D. none of these

Answer

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) given in the question.

1. **Define the functions**:
\[ f(x) = x^2 + x g'(1) + g''(2) \]
\[ g(x) = f(1) \cdot x^2 + x f'(x) + f''(x) \]

2. **Differentiate \( f(x) \)**:
Differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(x g'(1)) + \frac{d}{dx}(g''(2)) \]
Since \( g'(1) \) and \( g''(2) \) are constants, we have:
\[ f'(x) = 2x + g'(1) \]

3. **Differentiate \( f'(x) \)**:
Now differentiate \( f'(x) \):
\[ f''(x) = \frac{d}{dx}(2x + g'(1)) = 2 \]

4. **Evaluate \( f(1) \)**:
Substitute \( x = 1 \) into \( f(x) \):
\[ f(1) = 1^2 + 1 \cdot g'(1) + g''(2) = 1 + g'(1) + g''(2) \]

5. **Substitute \( f(1) \) into \( g(x) \)**:
Now substitute \( f(1) \) into \( g(x) \):
\[ g(x) = (1 + g'(1) + g''(2)) \cdot x^2 + x f'(x) + f''(x) \]
Substitute \( f'(x) \) and \( f''(x) \):
\[ g(x) = (1 + g'(1) + g''(2)) \cdot x^2 + x(2x + g'(1)) + 2 \]
Simplifying this gives:
\[ g(x) = (1 + g'(1) + g''(2)) \cdot x^2 + 2x^2 + xg'(1) + 2 \]
Combine like terms:
\[ g(x) = (3 + g'(1) + g''(2)) \cdot x^2 + xg'(1) + 2 \]

6. **Differentiate \( g(x) \)**:
Differentiate \( g(x) \):
\[ g'(x) = \frac{d}{dx}((3 + g'(1) + g''(2)) \cdot x^2 + xg'(1) + 2) \]
This gives:
\[ g'(x) = 2(3 + g'(1) + g''(2)) \cdot x + g'(1) \]

7. **Evaluate \( g'(1) \)**:
Substitute \( x = 1 \):
\[ g'(1) = 2(3 + g'(1) + g''(2)) + g'(1) \]
Rearranging gives:
\[ g'(1) = 6 + 2g'(1) + 2g''(2) \]
Rearranging further:
\[ -g'(1) = 6 + 2g''(2) \implies g'(1) = -6 - 2g''(2) \]

8. **Evaluate \( g''(2) \)**:
Differentiate \( g'(x) \):
\[ g''(x) = 2(3 + g'(1) + g''(2)) \]
Substitute \( x = 2 \) into \( g''(x) \):
\[ g''(2) = 2(3 + g'(1) + g''(2)) \]

9. **Final evaluations**:
We can now evaluate the options:
- For option 1: \( f'(1) + f'(2) = 0 \)
- For option 2: \( g'(2) = g'(1) \)
- For option 3: \( g''(2) + f''(3) = 6 \)

After evaluating all the steps and substituting back into the equations, we find that:

- **Option 1 is correct**: \( f'(1) + f'(2) = 0 \)