The period of ` f(x) = | cos^5 (x//2)|` is

Options

A. `pi`
B. ` 2 pi`
C. ` 4 pi`
D. None

Answer

To find the period of the function \( f(x) = |\cos^5(x/2)| \), we will follow these steps:

### Step 1: Identify the basic period of the cosine function
The basic cosine function \( \cos(x) \) has a period of \( 2\pi \). This means that \( \cos(x + 2\pi) = \cos(x) \).

**Hint:** Remember that the period of the cosine function is \( 2\pi \).

### Step 2: Determine the effect of the transformation
In our function \( f(x) = |\cos^5(x/2)| \), the argument of the cosine function is \( x/2 \). To find the period of \( \cos(x/2) \), we need to determine how the period changes with this transformation.

The period of \( \cos(kx) \) is given by \( \frac{2\pi}{k} \). Here, \( k = \frac{1}{2} \), so the period of \( \cos(x/2) \) is:

\[ \text{Period of } \cos(x/2) = \frac{2\pi}{1/2} = 4\pi \]

**Hint:** When the argument of a function is scaled, the period is adjusted inversely by that scale.

### Step 3: Consider the absolute value
Next, we need to consider the effect of the absolute value. The function \( |\cos(x)| \) has a period of \( \pi \) because \( |\cos(x + \pi)| = |\cos(x)| \).

**Hint:** The absolute value function can change the period, so check if it reduces the period.

### Step 4: Combine the effects
Now, we have two transformations affecting the period:
1. The \( \cos(x/2) \) contributes a period of \( 4\pi \).
2. The absolute value \( |\cdot| \) contributes a period of \( \pi \).

The overall period of \( f(x) = |\cos^5(x/2)| \) will be the least common multiple (LCM) of these two periods:

\[ \text{LCM}(4\pi, \pi) = 4\pi \]

### Conclusion
Thus, the period of the function \( f(x) = |\cos^5(x/2)| \) is \( 4\pi \).

**Final Answer:** The period of \( f(x) = |\cos^5(x/2)| \) is \( 4\pi \).

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