Let `f(x)=3x^(2)+4xg'(1)+g''(2)`
and, `g(x)=2x^(2)+3xf'(2)+f''(3)" for all "x in R.` Then,

Options

A. `f'(1)=22+12f'(2)`
B. `g,(2)=44+12g'(1)`
C. `f''(3)+g''(2)=10`
D. all the above

Explanation

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

Given:
1. \( f(x) = 3x^2 + 4x g'(1) + g''(2) \)
2. \( g(x) = 2x^2 + 3x f'(2) + f''(3) \)

### Step 1: Differentiate \( f(x) \)

We start by differentiating \( f(x) \):

\[ f'(x) = \frac{d}{dx}(3x^2 + 4x g'(1) + g''(2)) \]

Since \( g'(1) \) and \( g''(2) \) are constants, we differentiate:

\[ f'(x) = 6x + 4g'(1) \]

### Step 2: Differentiate \( g(x) \)

Next, we differentiate \( g(x) \):

\[ g'(x) = \frac{d}{dx}(2x^2 + 3x f'(2) + f''(3)) \]

Again, since \( f'(2) \) and \( f''(3) \) are constants, we differentiate:

\[ g'(x) = 4x + 3f'(2) \]

### Step 3: Evaluate \( f'(1) \) and \( g'(2) \)

Now we need to evaluate \( f'(1) \) and \( g'(2) \):

1. **For \( f'(1) \)**:

\[ f'(1) = 6(1) + 4g'(1) = 6 + 4g'(1) \]

2. **For \( g'(2) \)**:

\[ g'(2) = 4(2) + 3f'(2) = 8 + 3f'(2) \]

### Step 4: Substitute \( g'(1) \) and \( f'(2) \)

Now we substitute \( g'(1) \) from \( g'(x) \) evaluated at \( x=1 \):

\[ g'(1) = 4(1) + 3f'(2) = 4 + 3f'(2) \]

Substituting this into the equation for \( f'(1) \):

\[ f'(1) = 6 + 4(4 + 3f'(2)) = 6 + 16 + 12f'(2) = 22 + 12f'(2) \]

### Step 5: Substitute \( f'(2) \) into \( g'(2) \)

Now we can substitute \( f'(2) \) into the equation for \( g'(2) \):

\[ g'(2) = 8 + 3f'(2) \]

### Step 6: Evaluate \( f''(x) \) and \( g''(x) \)

Next, we differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = \frac{d}{dx}(6x + 4g'(1)) = 6 \]

And for \( g'(x) \):

\[ g''(x) = \frac{d}{dx}(4x + 3f'(2)) = 4 \]

### Step 7: Final Evaluation

Now we can evaluate \( f''(3) \) and \( g''(2) \):

- \( f''(3) = 6 \)
- \( g''(2) = 4 \)

### Conclusion

From our evaluations, we can summarize:

1. \( f'(1) = 22 + 12f'(2) \)
2. \( g'(2) = 8 + 3f'(2) \)
3. \( f''(3) = 6 \)
4. \( g''(2) = 4 \)

Thus, all options provided in the question are correct.