If `f(x)=max{tanx, sin x, cos x}` where `x in [-(pi)/(2),(3pi)/(2))` then the number of points, where f(x) is non -differentiable, is

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To solve the problem of finding the number of points where the function \( f(x) = \max\{\tan x, \sin x, \cos x\} \) is non-differentiable on the interval \( x \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right) \), we will follow these steps:

### Step 1: Identify the functions involved
The functions involved are:
- \( \tan x \)
- \( \sin x \)
- \( \cos x \)

### Step 2: Determine the intervals of each function
We need to analyze the behavior of each function within the given interval \( \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right) \).

- **For \( \tan x \)**: The function is undefined at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \). Within the interval, it will have a vertical asymptote at \( x = \frac{\pi}{2} \).

- **For \( \sin x \)**: This function is continuous and differentiable everywhere in the interval.

- **For \( \cos x \)**: This function is also continuous and differentiable everywhere in the interval.

### Step 3: Find points of intersection
Next, we need to find the points where these functions intersect, as the maximum function will switch between them at these points.

1. **Set \( \tan x = \sin x \)**:
- This occurs when \( \tan x = \sin x \) or \( \frac{\sin x}{\cos x} = \sin x \).
- This simplifies to \( \sin x (1 - \cos x) = 0 \).
- Solutions: \( x = 0 \) (since \( \sin x = 0 \) at \( x = 0, \pi, 2\pi, \ldots \))

2. **Set \( \tan x = \cos x \)**:
- This occurs when \( \tan x = \cos x \) or \( \frac{\sin x}{\cos x} = \cos x \).
- This simplifies to \( \sin x = \cos^2 x \).
- This equation can be solved numerically or graphically.

3. **Set \( \sin x = \cos x \)**:
- This occurs when \( \tan x = 1 \).
- Solutions: \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

### Step 4: Analyze non-differentiability
The function \( f(x) \) will be non-differentiable at points where the maximum function switches from one function to another. We need to check the points found in the previous step:

- **At \( x = 0 \)**: \( f(x) \) switches from \( \tan x \) to \( \sin x \).
- **At \( x = \frac{\pi}{4} \)**: \( f(x) \) switches from \( \sin x \) to \( \cos x \).
- **At \( x = \frac{\pi}{2} \)**: \( f(x) \) is non-differentiable because \( \tan x \) is undefined.
- **At \( x = \frac{5\pi}{4} \)**: \( f(x) \) switches from \( \cos x \) to \( \tan x \).

### Conclusion
The total points of non-differentiability are:
1. \( x = 0 \)
2. \( x = \frac{\pi}{4} \)
3. \( x = \frac{\pi}{2} \)
4. \( x = \frac{5\pi}{4} \)

Thus, the total number of points where \( f(x) \) is non-differentiable is **4**.