In `(0, 2pi)`, the total number of points where f(x)=max.{sin x, cos x, 1 - cos x} is not differentiable , are equal to

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the total number of points where the function \( f(x) = \max\{\sin x, \cos x, 1 - \cos x\} \) is not differentiable in the interval \( (0, 2\pi) \), we will follow these steps:

### Step 1: Analyze the functions
We need to analyze the three functions: \( \sin x \), \( \cos x \), and \( 1 - \cos x \) over the interval \( (0, 2\pi) \).

### Step 2: Find the intersection points
To determine where \( f(x) \) changes from one function to another, we need to find the points where these functions intersect.

1. **Set \( \sin x = \cos x \)**:
\[ \tan x = 1 \implies x = \frac{\pi}{4} + n\pi \]
In the interval \( (0, 2\pi) \), the solutions are \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

2. **Set \( \sin x = 1 - \cos x \)**:
\[ \sin x + \cos x = 1 \]
This can be solved using the identity \( \sin x = \sqrt{1 - \cos^2 x} \). Squaring both sides gives:
\[ 1 - \cos^2 x + \cos^2 x = 1 \implies \sin x = 1 \implies x = \frac{\pi}{2} \]

3. **Set \( \cos x = 1 - \cos x \)**:
\[ 2\cos x = 1 \implies \cos x = \frac{1}{2} \]
This gives:
\[ x = \frac{\pi}{3}, \frac{5\pi}{3} \]

### Step 3: List all intersection points
From the above calculations, the intersection points in the interval \( (0, 2\pi) \) are:
- \( x = \frac{\pi}{4} \)
- \( x = \frac{5\pi}{4} \)
- \( x = \frac{\pi}{2} \)
- \( x = \frac{\pi}{3} \)
- \( x = \frac{5\pi}{3} \)

### Step 4: Check for non-differentiability
The function \( f(x) \) is not differentiable at points where the maximum function switches from one function to another. This occurs at the intersection points we found.

### Step 5: Count non-differentiable points
Now we need to check if these points are indeed where the function is not differentiable:
1. \( x = \frac{\pi}{4} \)
2. \( x = \frac{5\pi}{4} \)
3. \( x = \frac{\pi}{2} \)
4. \( x = \frac{\pi}{3} \)
5. \( x = \frac{5\pi}{3} \)

All these points are where the maximum function changes, indicating non-differentiability.

### Conclusion
The total number of points where \( f(x) \) is not differentiable in the interval \( (0, 2\pi) \) is **5**.

### Final Answer
The total number of points where \( f(x) \) is not differentiable is **5**.

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