Let `f: R to R` be defined by
`f(x) = |sin 4x| + |cos 4x|, x in R`, Then period of f is

Options

A. `pi/8`
B. `pi/4`
C. `pi/2`
D. `pi`

Answer

To find the period of the function \( f(x) = |\sin(4x)| + |\cos(4x)| \), we can follow these steps:

### Step 1: Identify the individual periods of the components
The function consists of two parts: \( |\sin(4x)| \) and \( |\cos(4x)| \).

- The period of \( \sin(kx) \) is given by \( \frac{2\pi}{|k|} \).
- For \( \sin(4x) \), the period is:
\[ T_{\sin} = \frac{2\pi}{4} = \frac{\pi}{2} \]

- Similarly, the period of \( \cos(kx) \) is also \( \frac{2\pi}{|k|} \).
- For \( \cos(4x) \), the period is:
\[ T_{\cos} = \frac{2\pi}{4} = \frac{\pi}{2} \]

### Step 2: Determine the periods of the absolute values
The absolute value function does not change the period of the sine and cosine functions. Therefore, the periods of \( |\sin(4x)| \) and \( |\cos(4x)| \) remain the same:
- \( T_{|\sin(4x)|} = \frac{\pi}{2} \)
- \( T_{|\cos(4x)|} = \frac{\pi}{2} \)

### Step 3: Find the least common multiple (LCM) of the periods
Since both components have the same period, we can find the period of the entire function \( f(x) \) by taking the least common multiple (LCM) of the two periods:
\[ T_f = \text{lcm}\left(T_{|\sin(4x)|}, T_{|\cos(4x)|}\right) = \text{lcm}\left(\frac{\pi}{2}, \frac{\pi}{2}\right) = \frac{\pi}{2} \]

### Conclusion
Thus, the period of the function \( f(x) = |\sin(4x)| + |\cos(4x)| \) is:
\[ \boxed{\frac{\pi}{2}} \]