Let `f(x)=abs((x-1)) cos abs(x-2) sin abs(x-1)+abs(x-3) abs(x^2-5x+4)` . The number of points where the function is not differentiable is :

Options

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

Answer

To find the number of points where the function \( f(x) = |x-1| \cos |x-2| \sin |x-1| + |x-3| |x^2 - 5x + 4| \) is not differentiable, we will analyze the function step by step.

### Step 1: Identify points where the absolute value functions change
The absolute value functions \( |x-1| \), \( |x-2| \), and \( |x-3| \) can change their behavior at specific points. We need to find these points:
- \( |x-1| \) changes at \( x = 1 \)
- \( |x-2| \) changes at \( x = 2 \)
- \( |x-3| \) changes at \( x = 3 \)

Additionally, we need to consider the quadratic expression \( |x^2 - 5x + 4| \). Factoring gives us:
\[ x^2 - 5x + 4 = (x-1)(x-4) \]
This expression changes at \( x = 1 \) and \( x = 4 \).

### Step 2: List all critical points
From the analysis, the critical points where the function may not be differentiable are:
- From \( |x-1| \): \( x = 1 \)
- From \( |x-2| \): \( x = 2 \)
- From \( |x-3| \): \( x = 3 \)
- From \( |x^2 - 5x + 4| \): \( x = 1 \) and \( x = 4 \)

Thus, the critical points are \( x = 1, 2, 3, 4 \).

### Step 3: Determine differentiability at each critical point
We will check the differentiability at each of these points:
1. **At \( x = 1 \)**:
- The function \( |x-1| \) has a cusp at \( x = 1 \).

2. **At \( x = 2 \)**:
- The function \( \cos |x-2| \) is continuous and differentiable, but \( |x-2| \) has a cusp at \( x = 2 \).

3. **At \( x = 3 \)**:
- The function \( |x-3| \) has a cusp at \( x = 3 \).

4. **At \( x = 4 \)**:
- The function \( |x^2 - 5x + 4| \) has a cusp at \( x = 4 \).

### Step 4: Count the points of non-differentiability
The points where the function is not differentiable are:
- \( x = 1 \)
- \( x = 2 \)
- \( x = 3 \)
- \( x = 4 \)

Thus, there are a total of **4 points** where the function is not differentiable.

### Final Answer
The number of points where the function is not differentiable is **4**.

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