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 solve the problem, we need to analyze the function \( f(x) = \min\{|x^2 - 4|, |x^2 - 1|\} \) and determine the points where it is non-differentiable.

### Step-by-Step Solution

1. **Identify the Functions**:
The function consists of two parts:
- \( g_1(x) = |x^2 - 4| \)
- \( g_2(x) = |x^2 - 1| \)

2. **Find Critical Points**:
We need to find the points where each of these functions changes behavior, which occurs when the expressions inside the absolute values equal zero:
- For \( g_1(x) = |x^2 - 4| \):
\[ x^2 - 4 = 0 \implies x = \pm 2 \]
- For \( g_2(x) = |x^2 - 1| \):
\[ x^2 - 1 = 0 \implies x = \pm 1 \]

3. **Evaluate the Points**:
The critical points we found are \( x = -2, -1, 1, 2 \). These points are where the absolute value functions change from positive to negative or vice versa.

4. **Check for Non-Differentiability**:
The function \( f(x) \) will be non-differentiable at the points where the two functions intersect or where the individual functions change their behavior. We need to check the behavior at the critical points:
- At \( x = -2 \) and \( x = 2 \), \( g_1(x) \) changes from positive to negative.
- At \( x = -1 \) and \( x = 1 \), \( g_2(x) \) changes from positive to negative.

5. **Find Intersections**:
We also need to find where \( g_1(x) = g_2(x) \):
\[ |x^2 - 4| = |x^2 - 1| \]
This gives us two cases to solve:
- Case 1: \( x^2 - 4 = x^2 - 1 \) (no solutions)
- Case 2: \( x^2 - 4 = -(x^2 - 1) \)
\[ x^2 - 4 = -x^2 + 1 \implies 2x^2 = 5 \implies x = \pm \sqrt{\frac{5}{2}} \]

6. **List All Non-Differentiable Points**:
The points where \( f(x) \) is non-differentiable are:
- \( x = -2 \)
- \( x = -1 \)
- \( x = 1 \)
- \( x = 2 \)
- \( x = -\sqrt{\frac{5}{2}} \)
- \( x = \sqrt{\frac{5}{2}} \)

7. **Count the Points**:
We have identified a total of 6 points where \( f(x) \) is non-differentiable.

### Final Answer
The number of points where \( f(x) \) is non-differentiable is **6**.