Number of point where function f(x) defined as `f:[0,2pi] rarrR,f(x)={{:(3-|cosx-(1)/(sqrt2)|",",|sinx|lt(1)/(sqrt2)),(2+|cosx+(1)/(sqrt2)|",",|sinx|ge(1)/(sqrt2)):}` is non differentiable is

Options

A. 2
B. 4
C. 6
D. 0

Explanation

To determine the number of points where the function \( f(x) \) is non-differentiable, we need to analyze the given piecewise function:

\[ f(x) = \begin{cases} 3 - | \cos x - \frac{1}{\sqrt{2}} | & \text{if } |\sin x| < \frac{1}{\sqrt{2}} \\ 2 + | \cos x + \frac{1}{\sqrt{2}} | & \text{if } |\sin x| \geq \frac{1}{\sqrt{2}} \end{cases} \]

### Step 1: Identify the conditions for the piecewise function
The function is defined in two cases based on the value of \( |\sin x| \):
1. Case 1: \( |\sin x| < \frac{1}{\sqrt{2}} \)
2. Case 2: \( |\sin x| \geq \frac{1}{\sqrt{2}} \)

### Step 2: Find the critical points for \( |\sin x| = \frac{1}{\sqrt{2}} \)
The points where \( |\sin x| = \frac{1}{\sqrt{2}} \) occur at:
- \( \sin x = \frac{1}{\sqrt{2}} \) at \( x = \frac{\pi}{4}, \frac{3\pi}{4} \)
- \( \sin x = -\frac{1}{\sqrt{2}} \) at \( x = \frac{5\pi}{4}, \frac{7\pi}{4} \)

Thus, the critical points are:
- \( x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \)

### Step 3: Check differentiability at the critical points
At each of these points, we need to check if the function is non-differentiable. This can happen if:
- The function is not continuous at these points.
- The left-hand derivative and right-hand derivative at these points do not match.

### Step 4: Evaluate the function at the critical points
1. **At \( x = \frac{\pi}{4} \)**:
- \( |\sin x| < \frac{1}{\sqrt{2}} \): Use \( f(x) = 3 - |\cos x - \frac{1}{\sqrt{2}}| \)
- \( |\sin x| \geq \frac{1}{\sqrt{2}} \): Use \( f(x) = 2 + |\cos x + \frac{1}{\sqrt{2}}| \)

2. **At \( x = \frac{3\pi}{4} \)**:
- Similar evaluation as above.

3. **At \( x = \frac{5\pi}{4} \)**:
- Similar evaluation as above.

4. **At \( x = \frac{7\pi}{4} \)**:
- Similar evaluation as above.

### Step 5: Conclusion
Since the function switches between two different expressions at these four points, it will be non-differentiable at each of these points.

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

### Final Answer:
The number of points where the function \( f(x) \) is non-differentiable is **4**.

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