Consider the following function for the next two items that follow:
`f(x) = {{:(3x^(2) + 12x -1, -1 le x le 2),(37-x, 2 lt x le 3):}`
Which of the following 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

Answer

To solve the problem, we need to analyze the piecewise function given and determine whether the statements about its increasing and decreasing behavior are correct.

### Step 1: Define the function
The function is given in piecewise form:
- For the interval \([-1, 2]\):
\[ f(x) = 3x^2 + 12x - 1 \]
- For the interval \((2, 3]\):
\[ f(x) = 37 - x \]

### Step 2: Differentiate the function
We need to find the derivative \(f'(x)\) to determine where the function is increasing or decreasing.

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

2. For the interval \((2, 3]\):
\[ f'(x) = \frac{d}{dx}(37 - x) = -1 \]

### Step 3: Analyze the derivative in the intervals
1. **Interval \([-1, 2]\)**:
- We check the sign of \(f'(x) = 6x + 12\).
- At \(x = -1\):
\[ f'(-1) = 6(-1) + 12 = 6 \quad (\text{positive}) \]
- At \(x = 2\):
\[ f'(2) = 6(2) + 12 = 24 \quad (\text{positive}) \]
- Since \(f'(x)\) is positive for all \(x\) in \([-1, 2]\), the function is increasing in this interval.

2. **Interval \((2, 3]\)**:
- Here, \(f'(x) = -1\) which is negative.
- This indicates that the function is decreasing in the interval \((2, 3]\).

### Step 4: Conclusion
Based on the analysis:
1. The function \(f(x)\) is **increasing** in the interval \([-1, 2]\).
2. The function \(f(x)\) is **decreasing** in the interval \((2, 3]\).

Thus, both statements provided in the question are correct.

### Final Answer
Both statements are correct.