Find the fundamental period of the following function:
`f(x)="cos"3/5 x-"sin"2/7x`.

Explanation

To find the fundamental period of the function \( f(x) = \cos\left(\frac{3}{5}x\right) - \sin\left(\frac{2}{7}x\right) \), we will follow these steps:

### Step 1: Identify the periods of each function
The fundamental period of a cosine function \( \cos(kx) \) is given by:
\[ P = \frac{2\pi}{k} \]
Similarly, for a sine function \( \sin(kx) \), the period is also:
\[ P = \frac{2\pi}{k} \]

For our function:
- For \( \cos\left(\frac{3}{5}x\right) \):
\[ P_1 = \frac{2\pi}{\frac{3}{5}} = 2\pi \cdot \frac{5}{3} = \frac{10\pi}{3} \]

- For \( \sin\left(\frac{2}{7}x\right) \):
\[ P_2 = \frac{2\pi}{\frac{2}{7}} = 2\pi \cdot \frac{7}{2} = 7\pi \]

### Step 2: Find the LCM of the periods
The fundamental period of the combined function is the least common multiple (LCM) of the individual periods \( P_1 \) and \( P_2 \).

We have:
- \( P_1 = \frac{10\pi}{3} \)
- \( P_2 = 7\pi \)

To find the LCM, we can express both periods in terms of a common denominator. The LCM of the numerators will be calculated, and the HCF of the denominators will be taken.

1. Convert \( P_1 \) to have a common denominator with \( P_2 \):
\[ P_1 = \frac{10\pi}{3} = \frac{10\pi}{3} \cdot \frac{7}{7} = \frac{70\pi}{21} \]
\[ P_2 = 7\pi = \frac{7\pi}{1} \cdot \frac{21}{21} = \frac{147\pi}{21} \]

2. Now we have:
- \( P_1 = \frac{70\pi}{21} \)
- \( P_2 = \frac{147\pi}{21} \)

3. The LCM of the numerators \( 70 \) and \( 147 \):
- The prime factorization of \( 70 = 2 \times 5 \times 7 \)
- The prime factorization of \( 147 = 3 \times 7^2 \)

The LCM is obtained by taking the highest power of each prime:
\[ \text{LCM}(70, 147) = 2^1 \times 3^1 \times 5^1 \times 7^2 = 2 \times 3 \times 5 \times 49 = 1470 \]

### Step 3: Combine to find the fundamental period
The fundamental period \( T \) is given by:
\[ T = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{1470\pi}{21} = 70\pi \]

Thus, the fundamental period of the function \( f(x) = \cos\left(\frac{3}{5}x\right) - \sin\left(\frac{2}{7}x\right) \) is:
\[ \boxed{70\pi} \]