The function `f(x)=|sinx|`

Options

A. f is differentiable everywhere
B. f is continuous everywhere but not differentiable at `x=npi, ninZ`
C. f is continuous everywhere but not differentiable at `x=(2n+1)(pi)/(2), n in Z`
D. none of these

Answer

To determine the properties of the function \( f(x) = |\sin x| \), we will analyze its continuity and differentiability step by step.

### Step 1: Understanding the Function
The function \( f(x) = |\sin x| \) takes the sine function and applies the absolute value to it. This means that wherever the sine function is negative, the output of \( f(x) \) will be positive.

### Step 2: Analyzing Continuity
The sine function \( \sin x \) is continuous everywhere. Since the absolute value function is also continuous, the composition of these two functions, \( f(x) = |\sin x| \), is continuous everywhere.

### Step 3: Points of Non-Differentiability
To find where \( f(x) \) is not differentiable, we need to look for points where the sine function crosses zero, as these are the points where the absolute value function changes from positive to negative or vice versa.

The sine function is zero at:
- \( x = n\pi \) where \( n \) is any integer (i.e., \( n \in \mathbb{Z} \)).

At these points, the function \( f(x) \) has a sharp corner (the slope changes abruptly), which means it is not differentiable at these points.

### Step 4: Conclusion
Based on the analysis:
- The function \( f(x) = |\sin x| \) is continuous everywhere.
- The function is not differentiable at \( x = n\pi \) where \( n \in \mathbb{Z} \).

Thus, the correct option is:
**f is continuous everywhere but not differentiable at \( x = n\pi \) where \( n \in \mathbb{Z} \).**