Let `f:RtoR` be a function such that `f(x)=x^(3)+x^(2)f'(1)+xf''(2)+f'''(3)` for `x in R`
Consider the following
1. `f(2)=f(1)-f(0)`
`2.f''(2)-2f'(1)=12`
Which of the above is/are correct ?

Options

A. Only 1
B. Only 2
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to analyze the function given and the two statements provided.

### Step 1: Understanding the function
The function is defined as:
\[ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \]

### Step 2: Finding the derivatives
We will find the first, second, and third derivatives of \( f(x) \).

1. **First derivative \( f'(x) \)**:
\[ f'(x) = 3x^2 + 2x f'(1) + f''(2) \]
Here, \( f''(2) \) is constant with respect to \( x \).

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

3. **Third derivative \( f'''(x) \)**:
\[ f'''(x) = 6 \]
This is constant.

### Step 3: Evaluating the function at specific points
Now we will evaluate \( f(0) \), \( f(1) \), and \( f(2) \).

1. **Calculating \( f(0) \)**:
\[ f(0) = 0^3 + 0^2 f'(1) + 0 \cdot f''(2) + f'''(3) = f'''(3) = 6 \]

2. **Calculating \( f(1) \)**:
\[ f(1) = 1^3 + 1^2 f'(1) + 1 \cdot f''(2) + f'''(3) = 1 + f'(1) + f''(2) + 6 \]
Thus,
\[ f(1) = 7 + f'(1) + f''(2) \]

3. **Calculating \( f(2) \)**:
\[ f(2) = 2^3 + 2^2 f'(1) + 2 \cdot f''(2) + f'''(3) = 8 + 4f'(1) + 2f''(2) + 6 \]
Thus,
\[ f(2) = 14 + 4f'(1) + 2f''(2) \]

### Step 4: Analyzing the statements
1. **First statement**: \( f(2) = f(1) - f(0) \)
\[ 14 + 4f'(1) + 2f''(2) = (7 + f'(1) + f''(2)) - 6 \]
Simplifying the right side:
\[ 7 + f'(1) + f''(2) - 6 = 1 + f'(1) + f''(2) \]
Setting both sides equal:
\[ 14 + 4f'(1) + 2f''(2) = 1 + f'(1) + f''(2) \]
Rearranging gives:
\[ 13 + 3f'(1) + f''(2) = 0 \quad \text{(Equation 1)} \]

2. **Second statement**: \( f''(2) - 2f'(1) = 12 \)
From our earlier expression for \( f''(2) \):
\[ f''(2) = 6 + 2f'(1) \quad \text{(Equation 2)} \]
Substituting into the second statement gives:
\[ (6 + 2f'(1)) - 2f'(1) = 12 \]
Simplifying:
\[ 6 = 12 \quad \text{(False)} \]

### Conclusion
1. The first statement is true.
2. The second statement is false.

Thus, the correct answer is that only the first statement is correct.