Which of the following function is periodic ?

Options

A. `sgn(e^(-x)) x gt 0`
B. `|sin x| + sinx`
C. `min (4 cos x, |x|)`
D. `[x+(1)/(2)] + [x-(1)/(2)]+2[-x]`, [.] represents greatest integral function

Answer

To determine which of the given functions is periodic, we will analyze each function step by step.

### Step 1: Analyze the first function \( f(x) = \text{sgn}(e^{-x}) \) for \( x > 0 \)

1. **Understanding the function**: The signum function \( \text{sgn}(y) \) is defined as:
- \( \text{sgn}(y) = 1 \) if \( y > 0 \)
- \( \text{sgn}(y) = 0 \) if \( y = 0 \)
- \( \text{sgn}(y) = -1 \) if \( y < 0 \)

2. **Evaluate \( e^{-x} \)**: For \( x > 0 \), \( e^{-x} \) is always positive (since the exponential function is always positive). Therefore, \( \text{sgn}(e^{-x}) = 1 \).

3. **Conclusion**: Since \( f(x) = 1 \) for all \( x > 0 \), it is a constant function. Constant functions are periodic with any period, so this function is periodic.

### Step 2: Analyze the second function \( f(x) = |\sin x| + \sin x \)

1. **Understanding periodicity**: The sine function \( \sin x \) has a period of \( 2\pi \), and \( |\sin x| \) has a period of \( \pi \).

2. **Finding the least common multiple (LCM)**: The LCM of the periods \( \pi \) and \( 2\pi \) is \( 2\pi \).

3. **Conclusion**: Since the LCM is \( 2\pi \), the function \( f(x) = |\sin x| + \sin x \) is periodic with a period of \( 2\pi \).

### Step 3: Analyze the third function \( f(x) = \min(4\cos x, |x|) \)

1. **Understanding the components**: The function \( 4\cos x \) is periodic with a period of \( 2\pi \), while \( |x| \) is not periodic since it increases indefinitely as \( x \) moves away from zero.

2. **Behavior of the minimum function**: The minimum of a periodic function and a non-periodic function is not periodic because the non-periodic function dominates as \( |x| \) increases.

3. **Conclusion**: Therefore, \( f(x) = \min(4\cos x, |x|) \) is not periodic.

### Step 4: Analyze the fourth function \( f(x) = \lfloor x + \frac{1}{2} \rfloor + \lfloor x - \frac{1}{2} \rfloor + 2\lfloor -x \rfloor \)

1. **Understanding the greatest integer function**: The greatest integer function \( \lfloor x \rfloor \) is periodic with a period of \( 1 \).

2. **Sum of periodic functions**: The sum of periodic functions with the same period is also periodic. Here, all components are periodic with a period of \( 1 \).

3. **Conclusion**: Therefore, \( f(x) \) is periodic.

### Final Conclusion

The periodic functions among the given options are:
1. \( f(x) = \text{sgn}(e^{-x}) \) (periodic)
2. \( f(x) = |\sin x| + \sin x \) (periodic)
3. \( f(x) = \min(4\cos x, |x|) \) (not periodic)
4. \( f(x) = \lfloor x + \frac{1}{2} \rfloor + \lfloor x - \frac{1}{2} \rfloor + 2\lfloor -x \rfloor \) (periodic)

### Summary of Periodic Functions
- The periodic functions are:
- \( f(x) = \text{sgn}(e^{-x}) \)
- \( f(x) = |\sin x| + \sin x \)
- \( f(x) = \lfloor x + \frac{1}{2} \rfloor + \lfloor x - \frac{1}{2} \rfloor + 2\lfloor -x \rfloor \)