`f(x) = (x)/(sin x)` is

Options

A. increasing in (0, 1)
B. decreasing in (0, 1)
C. increasing in `(0, (1)/(2))` and decreasing in `((1)/(2), 1)`
D. none of these

Explanation

To determine the nature of the function \( f(x) = \frac{x}{\sin x} \), we will analyze its increasing and decreasing behavior by finding its derivative and examining its sign.

### Step 1: Find the derivative of \( f(x) \)

Using the quotient rule for differentiation, which states that if \( f(x) = \frac{u}{v} \), then

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

where \( u = x \) and \( v = \sin x \).

- The derivative of \( u \) is \( u' = 1 \).
- The derivative of \( v \) is \( v' = \cos x \).

Now applying the quotient rule:

\[ f'(x) = \frac{(1)(\sin x) - (x)(\cos x)}{(\sin x)^2} \]

This simplifies to:

\[ f'(x) = \frac{\sin x - x \cos x}{(\sin x)^2} \]

### Step 2: Analyze the sign of \( f'(x) \)

To determine where \( f(x) \) is increasing or decreasing, we need to find where \( f'(x) > 0 \) or \( f'(x) < 0 \).

1. The denominator \( (\sin x)^2 \) is always positive for \( x \neq n\pi \) (where \( n \) is an integer).
2. Therefore, we only need to analyze the numerator \( \sin x - x \cos x \).

### Step 3: Set the numerator to zero

To find critical points, we set the numerator to zero:

\[ \sin x - x \cos x = 0 \]

This can be rearranged to:

\[ \sin x = x \cos x \]

### Step 4: Analyze the function in the interval \( (0, \frac{\pi}{2}) \)

We will check the behavior of \( f'(x) \) in the interval \( (0, \frac{\pi}{2}) \):

- At \( x = 0 \):
\[ f'(0) = \frac{\sin(0) - 0 \cdot \cos(0)}{(\sin(0))^2} \text{ (undefined, but we can analyze limits)} \]

- As \( x \) approaches \( 0 \), \( f'(x) \) approaches \( 1 \) (since \( \sin x \approx x \) and \( \cos x \approx 1 \)).

- At \( x = \frac{\pi}{2} \):
\[ f'(\frac{\pi}{2}) = \frac{\sin(\frac{\pi}{2}) - \frac{\pi}{2} \cdot \cos(\frac{\pi}{2})}{(\sin(\frac{\pi}{2}))^2} = \frac{1 - 0}{1} = 1 \]

### Step 5: Conclusion

Since \( f'(x) > 0 \) in the interval \( (0, \frac{\pi}{2}) \), we conclude that \( f(x) \) is increasing in this interval.

### Summary

- The function \( f(x) = \frac{x}{\sin x} \) is increasing in the interval \( (0, \frac{\pi}{2}) \).