Find the fundamental period of the following function:
`f(x)="sin" (pix)/4+"sin"(pix)/3`

Explanation

To find the fundamental period of the function \( f(x) = \sin\left(\frac{\pi x}{4}\right) + \sin\left(\frac{\pi x}{3}\right) \), we will follow these steps:

### Step 1: Determine the period of each sine function
The period \( P \) of the sine function \( \sin(kx) \) is given by the formula:
\[ P = \frac{2\pi}{k} \]

#### For the first term \( \sin\left(\frac{\pi x}{4}\right) \):
Here, \( k = \frac{\pi}{4} \).
\[ P_1 = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \cdot \frac{4}{\pi} = 8 \]

#### For the second term \( \sin\left(\frac{\pi x}{3}\right) \):
Here, \( k = \frac{\pi}{3} \).
\[ P_2 = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \cdot \frac{3}{\pi} = 6 \]

### Step 2: Find the least common multiple (LCM) of the periods
Now we need to find the least common multiple of \( P_1 \) and \( P_2 \):
\[ P_1 = 8, \quad P_2 = 6 \]

To find the LCM of 8 and 6, we can list the multiples:
- Multiples of 8: 8, 16, 24, 32, ...
- Multiples of 6: 6, 12, 18, 24, ...

The smallest common multiple is 24.

### Step 3: Conclusion
Thus, the fundamental period of the function \( f(x) \) is:
\[ \text{Fundamental Period} = 24 \]