The function `f(x)=sin""(pix)/(2)+2 cos ""(pix)/(3)-tan""(pix)/(4)` is periodic with period

Options

A. 6
B. 3
C. 4
D. 12

Answer

To determine the period of the function \( f(x) = \frac{1}{2} \sin(\pi x) + \frac{2}{3} \cos(\pi x) - \frac{1}{4} \tan(\pi x) \), we need to analyze the periods of each component of the function.

### Step 1: Identify the periods of the individual components

1. **Sine Function**: The period of \( \sin(kx) \) is given by \( \frac{2\pi}{k} \).
- For \( \sin(\pi x) \), \( k = \pi \).
- Therefore, the period \( T_1 \) is:
\[ T_1 = \frac{2\pi}{\pi} = 2 \]

2. **Cosine Function**: The period of \( \cos(kx) \) is also \( \frac{2\pi}{k} \).
- For \( \cos(\pi x) \), \( k = \pi \).
- Therefore, the period \( T_2 \) is:
\[ T_2 = \frac{2\pi}{\pi} = 2 \]

3. **Tangent Function**: The period of \( \tan(kx) \) is given by \( \frac{\pi}{k} \).
- For \( \tan(\pi x) \), \( k = \pi \).
- Therefore, the period \( T_3 \) is:
\[ T_3 = \frac{\pi}{\pi} = 1 \]

### Step 2: Find the least common multiple (LCM) of the periods

Now, we need to find the least common multiple of the periods \( T_1 \), \( T_2 \), and \( T_3 \):
- \( T_1 = 2 \)
- \( T_2 = 2 \)
- \( T_3 = 1 \)

The LCM of \( 2, 2, \) and \( 1 \) is \( 2 \).

### Step 3: Conclusion

Thus, the function \( f(x) \) is periodic with a period of \( 2 \).

### Final Answer:
The period of the function \( f(x) \) is \( 2 \).

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