The period of `f(x)=sin""(2pix)/(3)+ cos""(pix)/(2)`, is

Options

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

Explanation

To find the period of the function \( f(x) = \sin\left(\frac{2\pi}{3} x\right) + \cos\left(\frac{\pi}{2} x\right) \), we will follow these steps:

### Step 1: Identify the angular frequencies
The function consists of two parts: \( \sin\left(\frac{2\pi}{3} x\right) \) and \( \cos\left(\frac{\pi}{2} x\right) \).
- For \( \sin\left(\frac{2\pi}{3} x\right) \), the angular frequency \( \omega_1 = \frac{2\pi}{3} \).
- For \( \cos\left(\frac{\pi}{2} x\right) \), the angular frequency \( \omega_2 = \frac{\pi}{2} \).

### Step 2: Calculate the periods of each function
The period \( T \) of a function is given by the formula:
\[ T = \frac{2\pi}{\omega} \]
- For \( \sin\left(\frac{2\pi}{3} x\right) \):
\[ T_1 = \frac{2\pi}{\frac{2\pi}{3}} = 3 \]
- For \( \cos\left(\frac{\pi}{2} x\right) \):
\[ T_2 = \frac{2\pi}{\frac{\pi}{2}} = 4 \]

### Step 3: Find the least common multiple (LCM) of the periods
To find the period of the combined function \( f(x) \), we take the LCM of \( T_1 \) and \( T_2 \):
- \( T_1 = 3 \)
- \( T_2 = 4 \)

The LCM of 3 and 4 is 12.

### Conclusion
Thus, the period of the function \( f(x) = \sin\left(\frac{2\pi}{3} x\right) + \cos\left(\frac{\pi}{2} x\right) \) is **12**.

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