A function `f (x)= max (sin x, cos x,1- cos x)` is non-derivable for n values of `x in [0, 2pi].` Then the vaue of n is:

Options

A. 2
B. 1
C. 3
D. 4

Answer

To solve the problem, we need to analyze the function \( f(x) = \max(\sin x, \cos x, 1 - \cos x) \) over the interval \( [0, 2\pi] \) and determine the points where the function is non-derivable.

### Step-by-Step Solution:

1. **Identify the Functions**:
We have three functions to consider:
- \( \sin x \)
- \( \cos x \)
- \( 1 - \cos x \)

2. **Find the Intervals**:
We will analyze the behavior of each function over the interval \( [0, 2\pi] \).
- \( \sin x \) starts at 0, reaches 1 at \( \frac{\pi}{2} \), goes back to 0 at \( \pi \), reaches -1 at \( \frac{3\pi}{2} \), and returns to 0 at \( 2\pi \).
- \( \cos x \) starts at 1, goes to 0 at \( \frac{\pi}{2} \), reaches -1 at \( \pi \), goes back to 0 at \( \frac{3\pi}{2} \), and returns to 1 at \( 2\pi \).
- \( 1 - \cos x \) starts at 0, reaches 1 at \( \frac{\pi}{2} \), reaches 2 at \( \pi \), goes back to 1 at \( \frac{3\pi}{2} \), and returns to 0 at \( 2\pi \).

3. **Graph the Functions**:
We can sketch the graphs of \( \sin x \), \( \cos x \), and \( 1 - \cos x \) on the same axes to visualize where they intersect.

4. **Find Intersection Points**:
We need to find the points where any two of these functions are equal, as these points will be candidates for non-derivability:
- Set \( \sin x = \cos x \):
\[ \tan x = 1 \implies x = \frac{\pi}{4}, \frac{5\pi}{4} \]
- Set \( \sin x = 1 - \cos x \):
\[ \sin x + \cos x = 1 \implies \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = 1 \implies x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]
- Set \( \cos x = 1 - \cos x \):
\[ 2\cos x = 1 \implies \cos x = \frac{1}{2} \implies x = \frac{\pi}{3}, \frac{5\pi}{3} \]

5. **List All Intersection Points**:
From the above calculations, we have the following intersection points:
- \( x = \frac{\pi}{4} \)
- \( x = \frac{5\pi}{4} \)
- \( x = \frac{3\pi}{4} \)
- \( x = \frac{7\pi}{4} \)
- \( x = \frac{\pi}{3} \)
- \( x = \frac{5\pi}{3} \)

6. **Count Non-Differentiable Points**:
The function \( f(x) \) is non-derivable at the points where the maximum function switches from one function to another. After checking the intersections, we find that there are three distinct points where the maximum function changes:
- \( x = \frac{\pi}{4} \)
- \( x = \frac{3\pi}{4} \)
- \( x = \frac{5\pi}{4} \)

Thus, the value of \( n \) is **3**.

### Final Answer:
The value of \( n \) is **3**.