Number of points where the function `f(x)=(x^2-1)|x^2-x-2| + sin(|x|)` is not differentiable, is: (A) 0 (B) 1 (C) 2 (D) 3

Options

A. 0
B. 1
C. 2
D. 3

Answer

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

### Step 1: Identify the components of the function
The function can be broken down into two parts:
1. \( g(x) = (x^2 - 1) |x^2 - x - 2| \)
2. \( h(x) = \sin(|x|) \)

### Step 2: Analyze \( g(x) \)
First, we need to find where \( g(x) \) is not differentiable. The term \( |x^2 - x - 2| \) will affect the differentiability of \( g(x) \).

#### Step 2.1: Find the roots of \( x^2 - x - 2 \)
To find the points where \( |x^2 - x - 2| \) changes, we solve:
\[ x^2 - x - 2 = 0 \]
Factoring gives:
\[ (x - 2)(x + 1) = 0 \]
Thus, the roots are \( x = 2 \) and \( x = -1 \).

#### Step 2.2: Analyze the behavior around the roots
The expression \( |x^2 - x - 2| \) changes its sign at \( x = -1 \) and \( x = 2 \). We need to check the differentiability at these points.

- For \( x < -1 \): \( x^2 - x - 2 > 0 \) so \( |x^2 - x - 2| = x^2 - x - 2 \)
- For \( -1 < x < 2 \): \( x^2 - x - 2 < 0 \) so \( |x^2 - x - 2| = -(x^2 - x - 2) = -x^2 + x + 2 \)
- For \( x > 2 \): \( x^2 - x - 2 > 0 \) so \( |x^2 - x - 2| = x^2 - x - 2 \)

At \( x = -1 \) and \( x = 2 \), \( g(x) \) has corners, which means it is not differentiable at these points.

### Step 3: Analyze \( h(x) \)
Next, we analyze \( h(x) = \sin(|x|) \).

#### Step 3.1: Check for non-differentiability
The function \( \sin(|x|) \) is differentiable everywhere except at \( x = 0 \) because of the absolute value function. At \( x = 0 \), there is a corner point.

### Step 4: Combine the results
From our analysis:
- \( g(x) \) is not differentiable at \( x = -1 \) and \( x = 2 \).
- \( h(x) \) is not differentiable at \( x = 0 \).

### Conclusion
The points where \( f(x) \) is not differentiable are:
1. \( x = -1 \)
2. \( x = 0 \)
3. \( x = 2 \)

Thus, the total number of points where the function is not differentiable is **3**.

### Final Answer
The number of points where the function \( f(x) \) is not differentiable is **3** (Option D).

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