Consider the function `f(x)=min{|x^(2)-9|,|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 \( f(x) = \min\{|x^2 - 9|, |x^2 - 1|\} \) is non-differentiable, we will analyze the points where the expressions inside the minimum function change their behavior.

### Step-by-Step Solution:

1. **Identify the critical points of the absolute values**:
- The expression \( |x^2 - 9| \) changes at \( x^2 - 9 = 0 \) which gives \( x = \pm 3 \).
- The expression \( |x^2 - 1| \) changes at \( x^2 - 1 = 0 \) which gives \( x = \pm 1 \).

2. **List all critical points**:
- From the above, the critical points are \( x = -3, -1, 1, 3 \).

3. **Evaluate \( f(x) \) at the critical points**:
- At \( x = -3 \):
\[ f(-3) = \min\{|(-3)^2 - 9|, |(-3)^2 - 1|\} = \min\{0, 8\} = 0 \]
- At \( x = -1 \):
\[ f(-1) = \min\{|(-1)^2 - 9|, |(-1)^2 - 1|\} = \min\{8, 0\} = 0 \]
- At \( x = 1 \):
\[ f(1) = \min\{|1^2 - 9|, |1^2 - 1|\} = \min\{8, 0\} = 0 \]
- At \( x = 3 \):
\[ f(3) = \min\{|3^2 - 9|, |3^2 - 1|\} = \min\{0, 8\} = 0 \]

4. **Determine the behavior of \( f(x) \) around the critical points**:
- The function \( f(x) \) is non-differentiable at points where the minimum function switches from one expression to another. This occurs at the critical points identified above.

5. **Check for additional non-differentiable points**:
- The points where the absolute values change (the vertices of the "V" shapes) are also non-differentiable. In this case, we have:
- The points \( x = -3, -1, 1, 3 \) are all points where \( f(x) \) is non-differentiable.

6. **Count the non-differentiable points**:
- The total number of non-differentiable points is \( 4 \) (from the critical points).

### Final Answer:
The number of points where \( f(x) \) is non-differentiable is **4**.