The period of the function
f (x) =|sin xl + |cos x| is

Options

A. `pi//2`
B. `pi`
C. `2 pi`
D. none of these

Answer

To find the period of the function \( f(x) = |\sin x| + |\cos x| \), we will follow these steps:

### Step 1: Understand the components of the function
The function consists of two parts: \( |\sin x| \) and \( |\cos x| \). Both of these functions have their own periods.

### Step 2: Determine the period of \( |\sin x| \)
The function \( \sin x \) has a period of \( 2\pi \). However, since we are taking the absolute value, \( |\sin x| \) has a period of \( \pi \). This is because \( |\sin(x + \pi)| = -\sin(x + \pi) = |\sin x| \).

### Step 3: Determine the period of \( |\cos x| \)
Similarly, the function \( \cos x \) also has a period of \( 2\pi \), and the absolute value \( |\cos x| \) has a period of \( \pi \) for the same reason as above.

### Step 4: Find the least common period
Since both \( |\sin x| \) and \( |\cos x| \) have a period of \( \pi \), we need to find the least common multiple (LCM) of their periods. The LCM of \( \pi \) and \( \pi \) is \( \pi \).

### Step 5: Verify the period
To confirm that \( f(x) = |\sin x| + |\cos x| \) has a period of \( \pi \), we check:
\[ f(x + \pi) = |\sin(x + \pi)| + |\cos(x + \pi)| \]
Using the properties of sine and cosine:
\[ |\sin(x + \pi)| = |\sin x| \quad \text{and} \quad |\cos(x + \pi)| = -|\cos x| = |\cos x| \]
Thus,
\[ f(x + \pi) = |\sin x| + |\cos x| = f(x) \]
This confirms that \( f(x) \) is periodic with period \( \pi \).

### Conclusion
The period of the function \( f(x) = |\sin x| + |\cos x| \) is \( \pi \).

---