The period of the function `|sin^3x/2|+|cos^5x/5|` is

Options

A. `2pi`
B. `10pi`
C. `8pi`
D. `5pi`

Answer

To find the period of the function \( f(x) = \left| \frac{\sin^3 x}{2} \right| + \left| \frac{\cos^5 x}{5} \right| \), we will analyze the periods of the individual components of the function.

### Step 1: Determine the period of \( \left| \frac{\sin^3 x}{2} \right| \)

1. The function \( \sin^3 x \) has a period of \( 2\pi \) because the sine function itself has a period of \( 2\pi \).
2. The period of \( \frac{\sin^3 x}{2} \) remains \( 2\pi \).
3. When we take the absolute value, \( \left| \sin^3 x \right| \), the period remains \( 2\pi \) because the absolute value does not change the periodicity of the sine function.
4. Thus, the period of \( \left| \frac{\sin^3 x}{2} \right| \) is \( 2\pi \).

### Step 2: Determine the period of \( \left| \frac{\cos^5 x}{5} \right| \)

1. The function \( \cos^5 x \) has a period of \( 2\pi \) as well because the cosine function has a period of \( 2\pi \).
2. The period of \( \frac{\cos^5 x}{5} \) remains \( 2\pi \).
3. Similar to the sine function, taking the absolute value does not change the periodicity. Therefore, \( \left| \cos^5 x \right| \) also has a period of \( 2\pi \).
4. Thus, the period of \( \left| \frac{\cos^5 x}{5} \right| \) is \( 2\pi \).

### Step 3: Combine the periods

1. Now we have the periods of both components:
- Period of \( \left| \frac{\sin^3 x}{2} \right| \) is \( 2\pi \).
- Period of \( \left| \frac{\cos^5 x}{5} \right| \) is \( 2\pi \).

2. Since both functions have the same period, the period of the entire function \( f(x) \) is the least common multiple (LCM) of their periods.
3. Therefore, the period of \( f(x) \) is:
\[ \text{Period of } f(x) = \text{LCM}(2\pi, 2\pi) = 2\pi \]

### Final Step: Conclusion

The period of the function \( f(x) = \left| \frac{\sin^3 x}{2} \right| + \left| \frac{\cos^5 x}{5} \right| \) is \( 2\pi \).