If `(dy)/(dx)=sin((x pi)/(2))cos(x pi)`, then y is strictly increasing in :

Options

A. (3, 4)
B. `(( 5)/(2), (7)/(2))`
C. (2, 3)
D. `((1)/(2), (3)/(2))`

Answer

To solve the problem, we need to determine the intervals where the function \( y \) is strictly increasing based on the given derivative \( \frac{dy}{dx} = \sin\left(\frac{x \pi}{2}\right) \cos(x \pi) \).

### Step-by-Step Solution:

1. **Understand the Condition for Increasing Function**:
A function \( y \) is strictly increasing where its derivative \( \frac{dy}{dx} > 0 \).

2. **Set Up the Inequality**:
We need to find where:
\[ \sin\left(\frac{x \pi}{2}\right) \cos(x \pi) > 0 \]

3. **Analyze Each Factor**:
- **Factor 1**: \( \sin\left(\frac{x \pi}{2}\right) > 0 \)
- **Factor 2**: \( \cos(x \pi) > 0 \)

4. **Determine Intervals for \( \sin\left(\frac{x \pi}{2}\right) > 0 \)**:
The sine function is positive in the intervals:
\[ n < x < n + 2 \quad \text{for } n \text{ even integers} \]
Specifically, this means:
- For \( n = 0 \): \( 0 < x < 2 \)
- For \( n = 2 \): \( 2 < x < 4 \)
- For \( n = 4 \): \( 4 < x < 6 \)
- And so on...

5. **Determine Intervals for \( \cos(x \pi) > 0 \)**:
The cosine function is positive in the intervals:
\[ 2k < x < 2k + 1 \quad \text{for } k \text{ integers} \]
Specifically, this means:
- For \( k = 0 \): \( 0 < x < 1 \)
- For \( k = 1 \): \( 2 < x < 3 \)
- For \( k = 2 \): \( 4 < x < 5 \)
- And so on...

6. **Find the Intersection of Intervals**:
We need to find the intersection of the intervals where both sine and cosine are positive:
- From \( \sin\left(\frac{x \pi}{2}\right) > 0 \): \( 0 < x < 2 \), \( 2 < x < 4 \)
- From \( \cos(x \pi) > 0 \): \( 0 < x < 1 \), \( 2 < x < 3 \)

The intersections are:
- For \( 0 < x < 2 \): The intersection is \( 0 < x < 1 \).
- For \( 2 < x < 4 \): The intersection is \( 2 < x < 3 \).

7. **Conclusion**:
The function \( y \) is strictly increasing in the intervals \( (0, 1) \) and \( (2, 3) \).

### Final Answer:
Thus, the correct option for which \( y \) is strictly increasing is:
- Option 2: \( \left(\frac{5}{2}, \frac{7}{2}\right) \) which corresponds to \( (2, 3) \).