Question

Let \[ f(x) = |\sin x| + |\cos x|, \quad x \in \mathbb{R}. \] The period of \( f(x) \) is:

• A. \( 2\pi \)
• B. \( \pi \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{2} \)
• E. \( \frac{3\pi}{2} \)

Hint:

When dealing with absolute value functions of periodic functions, analyze their behavior on one period and observe if the period is halved or modified due to the absolute values. In this case, the absolute values reduced the period.

Answer 1

The Correct Option is D

We are given the function: \[ f(x) = |\sin x| + |\cos x|. \] We need to find the period of this function.
Step 1: To determine the period of \( f(x) \), we need to analyze the behavior of \( |\sin x| \) and \( |\cos x| \). Both \( \sin x \) and \( \cos x \) have a period of \( 2\pi \), but since they are both taken as absolute values, the function \( f(x) \) will have a shorter period.
Step 2: Consider the behavior of \( f(x) \) in the interval \( [0, \pi] \):
- On the interval \( [0, \frac{\pi}{2}] \), both \( \sin x \) and \( \cos x \) are non-negative, so \( f(x) = \sin x + \cos x \).
- On the interval \( [\frac{\pi}{2}, \pi] \), \( \sin x \) is positive, but \( \cos x \) is negative, so \( f(x) = \sin x - \cos x \).
- The function \( f(x) \) repeats this pattern on subsequent intervals of length \( \pi \).
Step 3: Therefore, the function \( f(x) \) has a period of \( \frac{\pi}{2} \).
Thus, the period of \( f(x) \) is \( \frac{\pi}{2} \).
Therefore, the correct answer is option (D).