The number of points where `f(x)=abs(|x|^(2)-3|x|+2)` is not differentiable is/are

Options

A. 5
B. 3
C. 4
D. 1

Answer

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

### Step 1: Simplify the Inner Function
First, we need to simplify the expression inside the outer absolute value. Let \( g(x) = |x|^2 - 3|x| + 2 \).

### Step 2: Identify Critical Points of \( g(x) \)
To find where \( g(x) \) is not differentiable, we need to find the points where \( g(x) = 0 \):
\[ |x|^2 - 3|x| + 2 = 0 \]
This is a quadratic equation in terms of \( |x| \). Let \( u = |x| \):
\[ u^2 - 3u + 2 = 0 \]
Factoring gives:
\[ (u - 1)(u - 2) = 0 \]
Thus, \( u = 1 \) or \( u = 2 \). This means:
\[ |x| = 1 \quad \text{or} \quad |x| = 2 \]
This leads to the critical points \( x = 1, -1, 2, -2 \).

### Step 3: Analyze \( g(x) \)
The function \( g(x) \) changes its behavior at the points where \( |x| = 1 \) and \( |x| = 2 \). We will evaluate the function \( g(x) \) at these points:
- For \( x = 1 \):
\[ g(1) = |1|^2 - 3|1| + 2 = 1 - 3 + 2 = 0 \]
- For \( x = -1 \):
\[ g(-1) = |-1|^2 - 3|-1| + 2 = 1 - 3 + 2 = 0 \]
- For \( x = 2 \):
\[ g(2) = |2|^2 - 3|2| + 2 = 4 - 6 + 2 = 0 \]
- For \( x = -2 \):
\[ g(-2) = |-2|^2 - 3|-2| + 2 = 4 - 6 + 2 = 0 \]

### Step 4: Identify Points of Non-Differentiability
Now we have four points where \( g(x) = 0 \): \( x = -2, -1, 1, 2 \). At these points, the function \( g(x) \) is not differentiable because the absolute value function creates sharp corners.

### Step 5: Consider the Outer Absolute Value
Next, we need to consider the outer absolute value \( f(x) = |g(x)| \). The function \( f(x) \) will also be non-differentiable at the points where \( g(x) = 0 \) since the absolute value function will create additional sharp points.

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

Thus, there are **4 points** where the function \( f(x) \) is not differentiable.

### Final Answer
The number of points where \( f(x) \) is not differentiable is **4**.