The number of values of `x` in `(0, 2pi)` where the function `f(x) = (tan x + cotx)/2-|(tan x-cotx)/2|` continuous but non-derivable :

Options

A. 3
B. 4
C. 0
D. 1

Answer

To find the number of values of \( x \) in the interval \( (0, 2\pi) \) where the function

\[ f(x) = \frac{\tan x + \cot x}{2} - \left| \frac{\tan x - \cot x}{2} \right| \]

is continuous but not differentiable, we will analyze the function step by step.

### Step 1: Understanding the function
The function \( f(x) \) consists of two parts: \( \frac{\tan x + \cot x}{2} \) and \( \left| \frac{\tan x - \cot x}{2} \right| \). We need to determine where the modulus function affects the differentiability of \( f(x) \).

### Step 2: Identify conditions for \( \tan x \) and \( \cot x \)
The two cases we need to consider are:
1. When \( \tan x \geq \cot x \)
2. When \( \tan x < \cot x \)

### Step 3: Case 1 - \( \tan x \geq \cot x \)
If \( \tan x \geq \cot x \), then:

\[ \left| \frac{\tan x - \cot x}{2} \right| = \frac{\tan x - \cot x}{2} \]

Thus, the function simplifies to:

\[ f(x) = \frac{\tan x + \cot x}{2} - \frac{\tan x - \cot x}{2} = \cot x \]

### Step 4: Case 2 - \( \tan x < \cot x \)
If \( \tan x < \cot x \), then:

\[ \left| \frac{\tan x - \cot x}{2} \right| = -\frac{\tan x - \cot x}{2} \]

Thus, the function simplifies to:

\[ f(x) = \frac{\tan x + \cot x}{2} + \frac{\tan x - \cot x}{2} = \tan x \]

### Step 5: Identify points of non-differentiability
The function \( f(x) \) will be continuous everywhere except at points where \( \tan x \) and \( \cot x \) are undefined, which occurs at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

The points where \( \tan x = \cot x \) occur at:

\[ \tan x = 1 \Rightarrow x = \frac{\pi}{4}, \quad \tan x = -1 \Rightarrow x = \frac{5\pi}{4} \]

### Step 6: Summary of non-differentiable points
The function \( f(x) \) is not differentiable at the following points in the interval \( (0, 2\pi) \):

1. \( x = \frac{\pi}{4} \)
2. \( x = \frac{3\pi}{4} \)
3. \( x = \frac{5\pi}{4} \)
4. \( x = \frac{7\pi}{4} \)

### Conclusion
Thus, there are **4 values of \( x \)** in the interval \( (0, 2\pi) \) where the function \( f(x) \) is continuous but not differentiable.