Consider the following statements in respect of the function f(x) = sinx :
1. f(x) increases in the interval `(0, pi)`.
2. f(x) decreases in the interval `((5pi)/(2), 3pi)`.
Which of the above statements is/are correct?

Options

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

Answer

To determine the correctness of the statements regarding the function \( f(x) = \sin x \), we will analyze the behavior of the sine function in the specified intervals.

### Step 1: Analyze the first statement
**Statement 1:** \( f(x) \) increases in the interval \( (0, \pi) \).

To check if \( f(x) = \sin x \) is increasing in the interval \( (0, \pi) \), we need to find the derivative of \( f(x) \).

\[ f'(x) = \cos x \]

Now, we evaluate the derivative in the interval \( (0, \pi) \):

- At \( x = 0 \), \( f'(0) = \cos(0) = 1 \) (positive)
- At \( x = \frac{\pi}{2} \), \( f'(\frac{\pi}{2}) = \cos(\frac{\pi}{2}) = 0 \)
- At \( x = \pi \), \( f'(\pi) = \cos(\pi) = -1 \) (negative)

Since \( f'(x) \) is positive in \( (0, \frac{\pi}{2}) \) and becomes zero at \( \frac{\pi}{2} \) and then negative in \( (\frac{\pi}{2}, \pi) \), we conclude that \( f(x) \) increases from \( 0 \) to \( \frac{\pi}{2} \) and then decreases from \( \frac{\pi}{2} \) to \( \pi \). Therefore, the statement that \( f(x) \) increases in the entire interval \( (0, \pi) \) is **incorrect**.

### Step 2: Analyze the second statement
**Statement 2:** \( f(x) \) decreases in the interval \( \left(\frac{5\pi}{2}, 3\pi\right) \).

Now, we check the behavior of \( f(x) \) in the interval \( \left(\frac{5\pi}{2}, 3\pi\right) \).

- The critical points of \( \sin x \) occur at \( \frac{5\pi}{2} \) and \( 3\pi \).
- At \( x = \frac{5\pi}{2} \), \( f'(\frac{5\pi}{2}) = \cos(\frac{5\pi}{2}) = 0 \) (point of inflection)
- At \( x = 3\pi \), \( f'(3\pi) = \cos(3\pi) = -1 \) (negative)

To determine the behavior in the interval \( \left(\frac{5\pi}{2}, 3\pi\right) \):

- In this interval, \( f'(x) \) is negative (since \( \cos x \) is negative for \( x \) in \( \left(\frac{5\pi}{2}, 3\pi\right) \)).
- Therefore, \( f(x) \) is indeed decreasing in this interval.

Thus, the second statement is **correct**.

### Conclusion
- Statement 1 is **incorrect**.
- Statement 2 is **correct**.

### Final Answer
Only statement 2 is correct.

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