The periodic function `f(x)=asinlamdax+bcoslamdax` is

Options

A. `(2pi)/(lambda )`
B. ` (pi)/(lambda )`
C. ` (2pi)/(|lambda|)`
D. `(pi)/(|lambda|)`

Explanation

To find the period of the function \( f(x) = a \sin(\lambda x) + b \cos(\lambda x) \), we can follow these steps:

### Step 1: Identify the periods of the sine and cosine functions
The period of the sine function \( \sin(x) \) is \( 2\pi \). Therefore, the period of \( \sin(\lambda x) \) can be calculated as follows:
\[ \text{Period of } \sin(\lambda x) = \frac{2\pi}{|\lambda|} \]

### Step 2: Calculate the period of the cosine function
Similarly, the period of the cosine function \( \cos(x) \) is also \( 2\pi \). Thus, the period of \( \cos(\lambda x) \) is:
\[ \text{Period of } \cos(\lambda x) = \frac{2\pi}{|\lambda|} \]

### Step 3: Determine the period of the combined function
Since both \( \sin(\lambda x) \) and \( \cos(\lambda x) \) have the same period \( \frac{2\pi}{|\lambda|} \), the period of the function \( f(x) = a \sin(\lambda x) + b \cos(\lambda x) \) is also:
\[ \text{Period of } f(x) = \frac{2\pi}{|\lambda|} \]

### Conclusion
Thus, the period of the function \( f(x) = a \sin(\lambda x) + b \cos(\lambda x) \) is:
\[ \text{Period} = \frac{2\pi}{|\lambda|} \]