f (x) is twice differentiable polynomial function such that f (1) = 1, f (2) = 4, f (3)=9, then

Options

A. `f'(x)=2, AA x in R`
B. There exists at least one x `in`(1,3) such that f"(x) = 2
C. There exists at least one x `in` (2,3) such that f'(x) = 5=f"(x)
D. There exists at least one x `in` (1,2) such that f"(x) = 3

Answer

To solve the problem, we need to determine the polynomial function \( f(x) \) based on the given conditions and then analyze the derivatives of this function.

### Step-by-Step Solution:

1. **Identify the form of the polynomial**:
Since \( f(x) \) is a twice differentiable polynomial function, we can assume it is a quadratic polynomial of the form:
\[ f(x) = ax^2 + bx + c \]

2. **Set up equations using the given values**:
We are given three conditions:
- \( f(1) = 1 \)
- \( f(2) = 4 \)
- \( f(3) = 9 \)

Using these conditions, we can set up the following equations:

- From \( f(1) = 1 \):
\[ a(1)^2 + b(1) + c = 1 \implies a + b + c = 1 \quad \text{(1)} \]

- From \( f(2) = 4 \):
\[ a(2)^2 + b(2) + c = 4 \implies 4a + 2b + c = 4 \quad \text{(2)} \]

- From \( f(3) = 9 \):
\[ a(3)^2 + b(3) + c = 9 \implies 9a + 3b + c = 9 \quad \text{(3)} \]

3. **Solve the system of equations**:
We can solve equations (1), (2), and (3) simultaneously.

- Subtract equation (1) from equation (2):
\[ (4a + 2b + c) - (a + b + c) = 4 - 1 \]
\[ 3a + b = 3 \quad \text{(4)} \]

- Subtract equation (2) from equation (3):
\[ (9a + 3b + c) - (4a + 2b + c) = 9 - 4 \]
\[ 5a + b = 5 \quad \text{(5)} \]

- Now, subtract equation (4) from equation (5):
\[ (5a + b) - (3a + b) = 5 - 3 \]
\[ 2a = 2 \implies a = 1 \]

- Substitute \( a = 1 \) back into equation (4):
\[ 3(1) + b = 3 \implies 3 + b = 3 \implies b = 0 \]

- Substitute \( a = 1 \) and \( b = 0 \) into equation (1):
\[ 1 + 0 + c = 1 \implies c = 0 \]

4. **Write the polynomial function**:
Thus, we have:
\[ f(x) = 1x^2 + 0x + 0 = x^2 \]

5. **Find the first and second derivatives**:
- The first derivative \( f'(x) \):
\[ f'(x) = 2x \]

- The second derivative \( f''(x) \):
\[ f''(x) = 2 \]

6. **Analyze the conditions**:
- Since \( f''(x) = 2 \) for all \( x \), it satisfies the condition that there exists at least one \( x \) in \( (1, 3) \) such that \( f''(x) = 2 \).
- For the first derivative \( f'(x) = 2x \), it is equal to 5 for \( x = 2.5 \), which lies in the interval \( (2, 3) \).
- The condition \( f''(x) = 3 \) is not satisfied since \( f''(x) = 2 \) for all \( x \).

### Conclusion:
The correct statement is that there exists at least one \( x \) in \( (1, 3) \) such that \( f''(x) = 2 \).