if f(x) be a twice differentiable function such that `f(x) =x^(2) " for " x=1,2,3,` then

Options

A. `f''(x)=2 " "AA x in [1,3]`
B. `f''(x) =2 " ""for some " x in (1,3)`
C. `f''(x) =2 " " AA x in (1,3)`
D. `f'(x) =2x " " AA x in (1,3)`

Answer

To solve the given problem, we need to analyze the function \( f(x) \) which is defined as \( f(x) = x^2 \) for \( x = 1, 2, 3 \). We also know that \( f(x) \) is a twice differentiable function. We will derive the first and second derivatives and check the validity of the given options.

### Step-by-Step Solution:

1. **Understanding the function**:
We know that \( f(x) = x^2 \) at the points \( x = 1, 2, 3 \).

2. **Finding the first derivative**:
We differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]
This derivative is valid for all \( x \), but we will check its behavior specifically at \( x = 1, 2, 3 \).

3. **Finding the second derivative**:
We differentiate \( f'(x) \):
\[ f''(x) = \frac{d}{dx}(2x) = 2 \]
This second derivative is also valid for all \( x \).

4. **Evaluating the derivatives at specific points**:
- At \( x = 1 \):
\[ f'(1) = 2 \cdot 1 = 2 \]
\[ f''(1) = 2 \]
- At \( x = 2 \):
\[ f'(2) = 2 \cdot 2 = 4 \]
\[ f''(2) = 2 \]
- At \( x = 3 \):
\[ f'(3) = 2 \cdot 3 = 6 \]
\[ f''(3) = 2 \]

5. **Analyzing the options**:
- **Option 1**: \( f'(x) = 2x \) for all \( x \) in the interval [1, 3]. This is true, but the function is not defined as \( f(x) = x^2 \) for all \( x \) in this interval. Thus, this option is incorrect.
- **Option 2**: \( f''(x) = 2 \) for some values of \( x \) in [1, 3]. This is true since \( f''(x) = 2 \) for all \( x \), including the values 1, 2, and 3. This option is correct.
- **Option 3**: \( f''(x) = 2 \) for all \( x \) in [1, 3]. This is true, but again, it is not defined for all \( x \) in this interval. Thus, this option is incorrect.
- **Option 4**: \( f'(x) = 2x \) for all \( x \) in [1, 3]. This is true, but it is not defined as \( f(x) = x^2 \) for all \( x \) in this interval. Thus, this option is incorrect.

### Conclusion:
The only correct option is **Option 2**: \( f''(x) = 2 \) for some values of \( x \) in [1, 3].