Let `f : RtoR` be a function such that `f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3),x in R`. Then f(2) equals

Options

A. 8
B. -2
C. -4
D. 30

Answer

To solve the problem, we need to find the value of \( f(2) \) given the function:

\[ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \]

### Step 1: Identify the derivatives at specific points
We need to find \( f'(1) \), \( f''(2) \), and \( f'''(3) \). We will use the function definition to derive these values.

### Step 2: Differentiate \( f(x) \)
First, we differentiate \( f(x) \):

1. **First Derivative**:
\[ f'(x) = 3x^2 + 2x f'(1) + f''(2) \]

2. **Second Derivative**:
\[ f''(x) = 6x + 2f'(1) \]

3. **Third Derivative**:
\[ f'''(x) = 6 \]

### Step 3: Substitute \( x = 1 \) into \( f'(x) \)
Now, substituting \( x = 1 \) into \( f'(x) \):
\[ f'(1) = 3(1)^2 + 2(1)f'(1) + f''(2) \]
This simplifies to:
\[ f'(1) = 3 + 2f'(1) + f''(2) \]
Rearranging gives us:
\[ f'(1) - 2f'(1) = 3 + f''(2) \implies -f'(1) = 3 + f''(2) \implies f'(1) + f''(2) = -3 \tag{1} \]

### Step 4: Substitute \( x = 2 \) into \( f''(x) \)
Next, we substitute \( x = 2 \) into \( f''(x) \):
\[ f''(2) = 6(2) + 2f'(1) = 12 + 2f'(1) \]
Substituting this into equation (1):
\[ f'(1) + (12 + 2f'(1)) = -3 \]
This simplifies to:
\[ f'(1) + 12 + 2f'(1) = -3 \implies 3f'(1) + 12 = -3 \implies 3f'(1) = -15 \implies f'(1) = -5 \]

### Step 5: Find \( f''(2) \)
Now substituting \( f'(1) = -5 \) back into the equation for \( f''(2) \):
\[ f''(2) = 12 + 2(-5) = 12 - 10 = 2 \]

### Step 6: Find \( f'''(3) \)
From our earlier calculation, we have:
\[ f'''(3) = 6 \]

### Step 7: Substitute values back into \( f(2) \)
Now we can substitute \( f'(1) \), \( f''(2) \), and \( f'''(3) \) back into the original function to find \( f(2) \):
\[ f(2) = 2^3 + 2^2(-5) + 2(2) + 6 \]
Calculating each term:
\[ = 8 - 20 + 4 + 6 \]
Combining these gives:
\[ = 8 + 4 + 6 - 20 = 18 - 20 = -2 \]

Thus, the value of \( f(2) \) is:
\[ \boxed{-2} \]