Which of the following statements is`//`are correct?

Options

A. x+ sinx is increasing function
B. sec x is neither increasing nor decreasing function
C. x + sin x is decreasing function
D. sec x is an increasing function

Answer

To determine which of the statements regarding the functions are correct, we will analyze the given functions step by step.

### Step 1: Analyze the function \( f(x) = x + \sin x \)

1. **Find the derivative**:
\[ f'(x) = \frac{d}{dx}(x + \sin x) = 1 + \cos x \]

2. **Determine the sign of the derivative**:
- The cosine function, \( \cos x \), varies between -1 and 1. Therefore:
\[ f'(x) = 1 + \cos x \geq 0 \quad \text{(since } 1 + (-1) = 0 \text{ and } 1 + 1 = 2\text{)} \]
- This means \( f'(x) \) is always greater than or equal to 0.

3. **Conclusion about monotonicity**:
- Since \( f'(x) \geq 0 \), the function \( f(x) = x + \sin x \) is an increasing function.

### Step 2: Analyze the function \( g(x) = \sec x \)

1. **Find the derivative**:
\[ g'(x) = \frac{d}{dx}(\sec x) = \sec x \tan x \]

2. **Determine the intervals**:
- The secant function, \( \sec x \), is positive in the intervals \( (0, \frac{\pi}{2}) \) and \( (\frac{3\pi}{2}, 2\pi) \).
- The tangent function, \( \tan x \), is positive in the intervals \( (0, \frac{\pi}{2}) \) and \( (\pi, \frac{3\pi}{2}) \).

3. **Analyze the derivative in different intervals**:
- **Interval \( (0, \frac{\pi}{2}) \)**: Both \( \sec x \) and \( \tan x \) are positive, so \( g'(x) > 0 \) (increasing).
- **Interval \( (\frac{\pi}{2}, \pi) \)**: \( \sec x \) is negative and \( \tan x \) is negative, so \( g'(x) > 0 \) (increasing).
- **Interval \( (\pi, \frac{3\pi}{2}) \)**: \( \sec x \) is negative and \( \tan x \) is positive, so \( g'(x) < 0 \) (decreasing).
- **Interval \( (\frac{3\pi}{2}, 2\pi) \)**: Both \( \sec x \) and \( \tan x \) are positive, so \( g'(x) > 0 \) (increasing).

4. **Conclusion about monotonicity**:
- The function \( g(x) = \sec x \) is neither entirely increasing nor decreasing over its entire domain.

### Final Conclusion:

- **Option 1**: Correct (since \( x + \sin x \) is increasing).
- **Option 2**: Correct (since \( \sec x \) is neither increasing nor decreasing).
- **Option 3**: Incorrect (since \( x + \sin x \) is not decreasing).
- **Option 4**: Incorrect (since \( \sec x \) is not always increasing).

### Summary of Correct Statements:
- The correct statements are Option 1 and Option 2.