Let `f(x) = |sinx| + |cosx|, g(x) = cos(cosx) + cos(sinx) ,h(x)={-x/2}+sinpix` , where { } representsfractional function, then the period of

Options

A. f(x) + g(x) is `pi`
B. f(x) - g(x) is `pi`
C. f(x) + g(x) + h(x) is `2pi`
D. f(x) + g(x) + h(x) is is non existent

Answer

To find the period of the functions \( f(x) = |\sin x| + |\cos x| \), \( g(x) = \cos(\cos x) + \cos(\sin x) \), and \( h(x) = \left\{ -\frac{x}{2} \right\} + \sin(\pi x) \), we will analyze each function step by step.

### Step 1: Determine the period of \( f(x) \)

1. **Function Definition**:
\[ f(x) = |\sin x| + |\cos x| \]

2. **Periodicity of \( |\sin x| \) and \( |\cos x| \)**:
- The function \( |\sin x| \) has a period of \( \pi \) because it repeats every half cycle of \( \sin x \).
- The function \( |\cos x| \) also has a period of \( \pi \).

3. **Finding the period of \( f(x) \)**:
- Since both components have the same period of \( \pi \), the period of \( f(x) \) is:
\[ t_1 = \pi \]

### Step 2: Determine the period of \( g(x) \)

1. **Function Definition**:
\[ g(x) = \cos(\cos x) + \cos(\sin x) \]

2. **Periodicity of \( \cos(\cos x) \)**:
- The function \( \cos x \) has a period of \( 2\pi \).
- Therefore, \( \cos(\cos x) \) will also have a period of \( 2\pi \).

3. **Periodicity of \( \cos(\sin x) \)**:
- Similarly, \( \sin x \) has a period of \( 2\pi \), so \( \cos(\sin x) \) will also have a period of \( 2\pi \).

4. **Finding the period of \( g(x) \)**:
- Both components have the same period of \( 2\pi \), thus:
\[ t_2 = 2\pi \]

### Step 3: Determine the period of \( h(x) \)

1. **Function Definition**:
\[ h(x) = \left\{ -\frac{x}{2} \right\} + \sin(\pi x) \]

2. **Periodicity of the fractional part function**:
- The fractional part function \( \left\{ x \right\} \) has a period of \( 1 \).
- The term \( -\frac{x}{2} \) does not affect the periodicity, so \( \left\{ -\frac{x}{2} \right\} \) also has a period of \( 1 \).

3. **Periodicity of \( \sin(\pi x) \)**:
- The function \( \sin(\pi x) \) has a period of \( 2 \) because:
\[ \text{Period} = \frac{2\pi}{\pi} = 2 \]

4. **Finding the period of \( h(x) \)**:
- The LCM of the periods \( 1 \) and \( 2 \) is:
\[ t_3 = 2 \]

### Step 4: Combine the periods to find the period of \( f(x) + g(x) + h(x) \)

1. **Finding the LCM**:
- We have:
\[ t_1 = \pi, \quad t_2 = 2\pi, \quad t_3 = 2 \]

2. **Finding LCM of \( t_1, t_2, t_3 \)**:
- The LCM of \( \pi \) and \( 2\pi \) is \( 2\pi \).
- The LCM of \( 2\pi \) and \( 2 \) involves considering the irrational number \( \pi \) and the rational number \( 2 \). The LCM of a rational and an irrational number does not exist.

### Conclusion

Thus, the period of \( f(x) + g(x) + h(x) \) does not exist.

### Final Answer:
The correct option is that the period of \( f(x) + g(x) + h(x) \) is non-existence.

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