`f(x)={{:(3x^(2)+12x-1,-1lexle2),(37-x",",2ltxle3):}`
Which of the following statements is /are correct?
1. f(x) is increasing in the interval [-1,2].
2. f(x) is decreasing in the interval (2,3].
Select the correct answer using the code given below:

Options

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

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined in two parts:

1. \( f(x) = 3x^2 + 12x - 1 \) for \( -1 \leq x \leq 2 \)
2. \( f(x) = 37 - x \) for \( 2 < x \leq 3 \)

We will check the two statements given in the question:

### Step 1: Analyze the first statement
**Statement 1:** \( f(x) \) is increasing in the interval \([-1, 2]\).

To determine if \( f(x) \) is increasing, we need to find the derivative \( f'(x) \) for the interval \([-1, 2]\).

1. **Calculate the derivative:**
\[ f'(x) = \frac{d}{dx}(3x^2 + 12x - 1) = 6x + 12 \]

2. **Check the sign of the derivative in the interval \([-1, 2]\):**
- Choose a test point, say \( x = 1 \):
\[ f'(1) = 6(1) + 12 = 18 \]
Since \( f'(1) > 0 \), the function is increasing at this point.

3. **Check the endpoints:**
- At \( x = -1 \):
\[ f'(-1) = 6(-1) + 12 = 6 > 0 \]
- At \( x = 2 \):
\[ f'(2) = 6(2) + 12 = 24 > 0 \]

Since \( f'(x) > 0 \) for all \( x \) in the interval \([-1, 2]\), we conclude that \( f(x) \) is increasing in this interval.

### Conclusion for Statement 1:
**Statement 1 is correct.**

### Step 2: Analyze the second statement
**Statement 2:** \( f(x) \) is decreasing in the interval \( (2, 3] \).

1. **For the interval \( (2, 3] \), the function is defined as:**
\[ f(x) = 37 - x \]

2. **Calculate the derivative:**
\[ f'(x) = \frac{d}{dx}(37 - x) = -1 \]

3. **Check the sign of the derivative:**
Since \( f'(x) = -1 < 0 \) for all \( x \) in the interval \( (2, 3] \), the function is decreasing in this interval.

### Conclusion for Statement 2:
**Statement 2 is correct.**

### Final Conclusion:
Both statements are correct.

### Summary of Steps:
1. Calculate the derivative of \( f(x) \) for the first interval and check its sign.
2. Confirm that \( f(x) \) is increasing in the interval \([-1, 2]\).
3. Calculate the derivative of \( f(x) \) for the second interval and check its sign.
4. Confirm that \( f(x) \) is decreasing in the interval \( (2, 3] \).