Consider the function `f(x)=min{|x^(2)-4|,|x^(2)-1|}`, then the number of points where f(x) is non - differentiable is/are

Options

A. 0
B. 7
C. 6
D. 4

Answer

To determine the number of points where the function \( f(x) = \min\{|x^2 - 4|, |x^2 - 1|\} \) is non-differentiable, we will analyze the points where the individual functions \( |x^2 - 4| \) and \( |x^2 - 1| \) change behavior.

### Step 1: Identify the critical points of each function

1. **For \( |x^2 - 4| \)**:
- The expression \( x^2 - 4 = 0 \) gives \( x^2 = 4 \) or \( x = \pm 2 \).
- The function \( |x^2 - 4| \) changes at \( x = -2 \) and \( x = 2 \).

2. **For \( |x^2 - 1| \)**:
- The expression \( x^2 - 1 = 0 \) gives \( x^2 = 1 \) or \( x = \pm 1 \).
- The function \( |x^2 - 1| \) changes at \( x = -1 \) and \( x = 1 \).

### Step 2: Identify the points of interest

The critical points where the functions change are:
- From \( |x^2 - 4| \): \( x = -2, 2 \)
- From \( |x^2 - 1| \): \( x = -1, 1 \)

Thus, the points of interest are \( -2, -1, 1, 2 \).

### Step 3: Analyze the behavior of \( f(x) \)

Now we need to analyze the behavior of \( f(x) \) around these critical points:

- **For \( x < -2 \)**:
- \( |x^2 - 4| = x^2 - 4 \) and \( |x^2 - 1| = x^2 - 1 \)
- So, \( f(x) = |x^2 - 4| \) since \( x^2 - 4 < x^2 - 1 \).

- **At \( x = -2 \)**:
- \( f(x) \) switches from \( |x^2 - 4| \) to \( |x^2 - 1| \). This is a potential point of non-differentiability.

- **For \( -2 < x < -1 \)**:
- \( |x^2 - 4| = x^2 - 4 \) and \( |x^2 - 1| = 1 - x^2 \)
- Here, \( f(x) = |x^2 - 1| \).

- **At \( x = -1 \)**:
- Another switch occurs, indicating another potential point of non-differentiability.

- **For \( -1 < x < 1 \)**:
- \( |x^2 - 4| = 4 - x^2 \) and \( |x^2 - 1| = 1 - x^2 \)
- Here, \( f(x) = |x^2 - 1| \).

- **At \( x = 1 \)**:
- Another switch occurs, indicating another potential point of non-differentiability.

- **For \( 1 < x < 2 \)**:
- \( |x^2 - 4| = 4 - x^2 \) and \( |x^2 - 1| = x^2 - 1 \)
- Here, \( f(x) = |x^2 - 4| \).

- **At \( x = 2 \)**:
- Another switch occurs, indicating another potential point of non-differentiability.

- **For \( x > 2 \)**:
- \( f(x) = |x^2 - 4| \).

### Step 4: Count the non-differentiable points

The points where \( f(x) \) is non-differentiable are:
1. \( x = -2 \)
2. \( x = -1 \)
3. \( x = 1 \)
4. \( x = 2 \)

In addition, we need to check the behavior at these points. The function is non-differentiable at each of these points due to the abrupt changes in the slopes of the graphs.

### Conclusion

Thus, the total number of points where \( f(x) \) is non-differentiable is **6**.