The period of f(x) = 3 sin`(pi x)/(3) + 4 cos ((pi x)/(4))` is

Options

A. 6
B. 8
C. 24
D. `2 pi`

Answer

To find the period of the function \( f(x) = 3 \sin\left(\frac{\pi x}{3}\right) + 4 \cos\left(\frac{\pi x}{4}\right) \), we need to determine the periods of the individual sine and cosine components and then find the least common multiple (LCM) of these periods.

### Step-by-Step Solution:

1. **Identify the components of the function**:
The function consists of two parts:
- \( 3 \sin\left(\frac{\pi x}{3}\right) \)
- \( 4 \cos\left(\frac{\pi x}{4}\right) \)

2. **Determine the period of the sine function**:
The general formula for the period of \( \sin(nx) \) is given by:
\[ \text{Period} = \frac{2\pi}{n} \]
For \( \sin\left(\frac{\pi x}{3}\right) \), we have \( n = \frac{\pi}{3} \). Thus, the period is:
\[ \text{Period of } \sin\left(\frac{\pi x}{3}\right) = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \cdot \frac{3}{\pi} = 6 \]

3. **Determine the period of the cosine function**:
Similarly, for the cosine function \( \cos(nx) \), the period is also given by:
\[ \text{Period} = \frac{2\pi}{n} \]
For \( \cos\left(\frac{\pi x}{4}\right) \), we have \( n = \frac{\pi}{4} \). Thus, the period is:
\[ \text{Period of } \cos\left(\frac{\pi x}{4}\right) = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \]

4. **Find the least common multiple (LCM) of the periods**:
Now we need to find the LCM of the two periods we calculated:
- Period of \( \sin\left(\frac{\pi x}{3}\right) = 6 \)
- Period of \( \cos\left(\frac{\pi x}{4}\right) = 8 \)

The multiples of 6 are: 6, 12, 18, 24, ...
The multiples of 8 are: 8, 16, 24, 32, ...

The smallest common multiple is 24.

5. **Conclusion**:
Therefore, the period of the function \( f(x) = 3 \sin\left(\frac{\pi x}{3}\right) + 4 \cos\left(\frac{\pi x}{4}\right) \) is:
\[ \text{Period} = 24 \]

### Final Answer:
The period of \( f(x) \) is **24**.