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 solve the problem, we need to analyze the function \( f(x) = \min\{|x^2 - 9|, |x^2 - 1|\} \) and determine the points where it is non-differentiable.

### Step-by-Step Solution:

1. **Identify the Functions**:
We have two functions to consider:
- \( g_1(x) = |x^2 - 9| \)
- \( g_2(x) = |x^2 - 1| \)

2. **Find Critical Points**:
The critical points for each function occur where the expressions inside the absolute values are zero:
- For \( g_1(x) = |x^2 - 9| \):
\[ x^2 - 9 = 0 \implies x^2 = 9 \implies x = \pm 3 \]
- For \( g_2(x) = |x^2 - 1| \):
\[ x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1 \]

3. **List the Critical Points**:
The critical points from both functions are:
\[ x = -3, -1, 1, 3 \]

4. **Determine the Behavior of \( f(x) \)**:
The function \( f(x) \) is defined as the minimum of the two absolute value functions. Therefore, we need to analyze the intervals defined by the critical points:
- \( (-\infty, -3) \)
- \( [-3, -1) \)
- \( [-1, 1) \)
- \( [1, 3) \)
- \( [3, \infty) \)

5. **Check for Non-Differentiability**:
The function \( f(x) \) will be non-differentiable at points where the minimum function switches from one function to another, or where the absolute value functions themselves are non-differentiable (i.e., at the critical points).

- At \( x = -3 \): \( g_1 \) switches from being the minimum to \( g_2 \).
- At \( x = -1 \): \( g_2 \) switches from being the minimum to \( g_1 \).
- At \( x = 1 \): \( g_2 \) switches from being the minimum to \( g_1 \).
- At \( x = 3 \): \( g_1 \) switches from being the minimum to \( g_2 \).

6. **Count the Points**:
The points where \( f(x) \) is non-differentiable are:
- \( x = -3 \)
- \( x = -1 \)
- \( x = 1 \)
- \( x = 3 \)

Additionally, we need to check the points where the absolute value functions themselves are non-differentiable:
- \( g_1(x) \) is non-differentiable at \( x = -3 \) and \( x = 3 \).
- \( g_2(x) \) is non-differentiable at \( x = -1 \) and \( x = 1 \).

Thus, the total points of non-differentiability are:
- \( x = -3 \)
- \( x = -1 \)
- \( x = 1 \)
- \( x = 3 \)

In total, we have **4 points** where \( f(x) \) is non-differentiable.

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