The period of the fuction
` f(x)=sin""((pix)/(n!))+ cos((pix)/((n+1)!))` , is

Options

A. `2xx(n+1)!`
B. `2(n!)`
C. `n+1`
D. none of these

Explanation

To find the period of the function

\[ f(x) = \sin\left(\frac{\pi x}{n!}\right) + \cos\left(\frac{\pi x}{(n+1)!}\right), \]

we will analyze the periods of the individual components of the function.

### Step 1: Determine the period of \(\sin\left(\frac{\pi x}{n!}\right)\)

The standard period of \(\sin(x)\) is \(2\pi\). When we have \(\sin(kx)\), the period becomes:

\[ \text{Period} = \frac{2\pi}{|k|}. \]

In our case, \(k = \frac{\pi}{n!}\). Therefore, the period of \(\sin\left(\frac{\pi x}{n!}\right)\) is:

\[ \text{Period} = \frac{2\pi}{\left|\frac{\pi}{n!}\right|} = \frac{2\pi \cdot n!}{\pi} = 2n!. \]

### Step 2: Determine the period of \(\cos\left(\frac{\pi x}{(n+1)!}\right)\)

Similarly, the standard period of \(\cos(x)\) is also \(2\pi\). For \(\cos(kx)\), the period is given by:

\[ \text{Period} = \frac{2\pi}{|k|}. \]

Here, \(k = \frac{\pi}{(n+1)!}\). Thus, the period of \(\cos\left(\frac{\pi x}{(n+1)!}\right)\) is:

\[ \text{Period} = \frac{2\pi}{\left|\frac{\pi}{(n+1)!}\right|} = \frac{2\pi \cdot (n+1)!}{\pi} = 2(n+1)!. \]

### Step 3: Find the least common multiple (LCM) of the two periods

Now, we need to find the least common multiple (LCM) of the two periods we calculated:

1. Period of \(\sin\left(\frac{\pi x}{n!}\right) = 2n!\)
2. Period of \(\cos\left(\frac{\pi x}{(n+1)!}\right) = 2(n+1)!\)

The LCM of \(2n!\) and \(2(n+1)!\) can be calculated as follows:

\[ \text{LCM}(2n!, 2(n+1)!) = 2 \cdot \text{LCM}(n!, (n+1)!). \]

Since \((n+1)! = (n+1) \cdot n!\), we have:

\[ \text{LCM}(n!, (n+1)!) = (n+1)!. \]

Thus,

\[ \text{LCM}(2n!, 2(n+1)!) = 2(n+1)!. \]

### Conclusion

The period of the function \(f(x)\) is:

\[ \text{Period of } f(x) = 2(n+1)!. \]