which of the following statement `is // are ` true ?
(i) f(x) =sin x is increasing in interval `[(-pi)/(2),(pi)/(2)]`
(ii) f(x) = sin x is increasing at all point of the interval `[(-pi)/(2),(pi)/(2)]`
(3) f(x) = sin x is increasing in interval `((-pi)/(2),(pi)/(2)) UU ((3pi)/(2),(5pi)/(2))`
(4) f(x)=sin x is increasing at all point of the interval `((-pi)/(2),(pi)/(2)) UU ((3pi)/(2),(5pi)/(2))`
(5) f(x) = sin x is increasing in intervals `[(-pi)/(2),(pi)/(2)]& [(3pi)/(2),(5pi)/(2)]`

Options

A. all are correct
B. all are false
C. (3) and (4) are correct
D. (1),(4) & (5) are correct

Answer

To determine which of the statements about the function \( f(x) = \sin x \) are true, we need to analyze the behavior of the sine function over the specified intervals. Here’s a step-by-step solution:

### Step 1: Analyze the function \( f(x) = \sin x \)
The sine function is periodic and oscillates between -1 and 1. Its derivative, \( f'(x) = \cos x \), tells us about its increasing and decreasing behavior.

### Step 2: Find the critical points
To find where \( f(x) \) is increasing or decreasing, we set the derivative to zero:
\[ f'(x) = \cos x = 0 \]
This occurs at:
\[ x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
The critical points in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) are \( x = \frac{\pi}{2} \).

### Step 3: Determine the sign of the derivative
- In the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\):
- \( f'(x) > 0 \) when \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)
- \( f'(x) = 0 \) at \( x = \frac{\pi}{2} \)

Thus, \( f(x) = \sin x \) is increasing on the interval \((- \frac{\pi}{2}, \frac{\pi}{2})\).

### Step 4: Analyze the interval \((\frac{3\pi}{2}, \frac{5\pi}{2})\)
- In the interval \((\frac{3\pi}{2}, \frac{5\pi}{2})\):
- \( f'(x) > 0 \) when \( x \in \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right) \)

Thus, \( f(x) = \sin x \) is also increasing on this interval.

### Step 5: Evaluate the statements
1. **Statement (i)**: \( f(x) = \sin x \) is increasing in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) - **True**.
2. **Statement (ii)**: \( f(x) = \sin x \) is increasing at all points of the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) - **True**.
3. **Statement (iii)**: \( f(x) = \sin x \) is increasing in the interval \((- \frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, \frac{5\pi}{2})\) - **True**.
4. **Statement (iv)**: \( f(x) = \sin x \) is increasing at all points in the interval \((- \frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, \frac{5\pi}{2})\) - **True**.
5. **Statement (v)**: \( f(x) = \sin x \) is increasing in intervals \([- \frac{\pi}{2}, \frac{\pi}{2}]\) & \((\frac{3\pi}{2}, \frac{5\pi}{2})\) - **True**.

### Conclusion
All statements (i), (ii), (iii), (iv), and (v) are true.