Consider the following fur the next two items that follow:
`f(x) = {{:(3x^(2) + 12x -1, -1 le x le 2),(37-x, 2 lt x le 3):}`
Winch of the following statements are correct?
1. f( x) is continuous at x = 2
2. f(x) attains greatest value at x = 2
3. f(x) is differentiate al x = 2.
Select the correct answer using the code given below:

Options

A. 1 and 2 only
B. 2 and 3 only
C. 1 and 3 only
D. 1,2 and 3

Answer

To solve the problem, we need to analyze the function \( f(x) \) given in two parts and check the three statements regarding its continuity, maximum value, and differentiability at \( x = 2 \).

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} 3x^2 + 12x - 1 & \text{for } -1 \leq x \leq 2 \\ 37 - x & \text{for } 2 < x \leq 3 \end{cases} \]

### Step 2: Check continuity at \( x = 2 \)
To check if \( f(x) \) is continuous at \( x = 2 \), we need to find the left-hand limit (LHL) and right-hand limit (RHL) at \( x = 2 \) and see if they are equal to \( f(2) \).

1. **Calculate \( f(2) \)**:
\[ f(2) = 3(2)^2 + 12(2) - 1 = 3(4) + 24 - 1 = 12 + 24 - 1 = 35 \]

2. **Calculate the left-hand limit as \( x \) approaches 2**:
\[ \text{LHL} = \lim_{x \to 2^-} f(x) = f(2) = 35 \]

3. **Calculate the right-hand limit as \( x \) approaches 2**:
\[ \text{RHL} = \lim_{x \to 2^+} f(x) = 37 - 2 = 35 \]

Since LHL = RHL = \( f(2) = 35 \), we conclude that \( f(x) \) is continuous at \( x = 2 \).

### Step 3: Check if \( f(x) \) attains the greatest value at \( x = 2 \)
1. The function \( f(x) = 3x^2 + 12x - 1 \) is a quadratic function that opens upwards (since the coefficient of \( x^2 \) is positive). It is increasing in the interval \([-1, 2]\).

2. The function \( f(x) = 37 - x \) is a linear function that is decreasing for \( x > 2 \).

Since \( f(x) \) is increasing up to \( x = 2 \) and decreasing thereafter, the maximum value occurs at \( x = 2 \).

### Step 4: Check differentiability at \( x = 2 \)
To check if \( f(x) \) is differentiable at \( x = 2 \), we need to compute the derivatives from both sides and see if they are equal.

1. **Differentiate \( f(x) \) for \( -1 \leq x \leq 2 \)**:
\[ f'(x) = \frac{d}{dx}(3x^2 + 12x - 1) = 6x + 12 \]
At \( x = 2 \):
\[ f'(2) = 6(2) + 12 = 12 + 12 = 24 \]

2. **Differentiate \( f(x) \) for \( 2 < x \leq 3 \)**:
\[ f'(x) = \frac{d}{dx}(37 - x) = -1 \]

Since \( f'(2) = 24 \) (from the left) and \( f'(2) = -1 \) (from the right), the derivatives do not match. Therefore, \( f(x) \) is not differentiable at \( x = 2 \).

### Conclusion
Based on our analysis:
1. \( f(x) \) is continuous at \( x = 2 \) (True).
2. \( f(x) \) attains the greatest value at \( x = 2 \) (True).
3. \( f(x) \) is not differentiable at \( x = 2 \) (False).

Thus, the correct statements are 1 and 2.

### Final Answer
The correct answer is option A: 1 and 2 only.