`f(x) = {:{(min{|x|,2-x^2}",",-2lexle2),([|x|]" ,",2 lt|x|le3):}`
where [x] denotes the greatest integer `le x` . The number of points , where f is not differentiable in (-3,3) is _______ .

Answer

To determine the number of points where the function \( f(x) \) is not differentiable in the interval \((-3, 3)\), we will analyze the function given:

\[ f(x) = \begin{cases} \min(|x|, 2 - x^2) & \text{for } -2 \leq x \leq 2 \\ \lfloor |x| \rfloor & \text{for } 2 < |x| \leq 3 \end{cases} \]

### Step 1: Analyze the first case \((-2 \leq x \leq 2)\)

In this interval, we have \( f(x) = \min(|x|, 2 - x^2) \).

1. **Find the points of intersection**:
- Set \( |x| = 2 - x^2 \).
- For \( x \geq 0 \): \( x = 2 - x^2 \) leads to \( x^2 + x - 2 = 0 \).
- The roots are \( x = 1 \) and \( x = -2 \).
- For \( x < 0 \): \( -x = 2 - x^2 \) leads to \( x^2 - x - 2 = 0 \).
- The roots are \( x = 2 \) and \( x = -1 \).

2. **Identify critical points**:
- The critical points from the intersection are \( x = -2, -1, 1, 2 \).

### Step 2: Check differentiability at critical points

- At \( x = -2 \):
- \( f(x) \) changes from \( 0 \) to \( 0 \) (not differentiable).

- At \( x = -1 \):
- \( f(x) \) changes from \( 1 \) to \( 0 \) (not differentiable).

- At \( x = 1 \):
- \( f(x) \) changes from \( 0 \) to \( 1 \) (not differentiable).

- At \( x = 2 \):
- \( f(x) \) changes from \( 0 \) to \( 2 \) (not differentiable).

### Step 3: Analyze the second case \((2 < |x| \leq 3)\)

In this interval, \( f(x) = \lfloor |x| \rfloor \).

- The values of \( |x| \) in this interval are:
- For \( 2 < x \leq 3 \): \( f(x) = 2 \) (constant).
- For \( -3 \leq x < -2 \): \( f(x) = 2 \) (constant).

### Step 4: Check differentiability at the boundaries

- At \( x = -2 \) (boundary between cases):
- \( f(x) \) changes from \( 0 \) to \( 2 \) (not differentiable).

- At \( x = 2 \) (boundary between cases):
- \( f(x) \) changes from \( 0 \) to \( 2 \) (not differentiable).

### Step 5: Summary of non-differentiable points

The points where \( f(x) \) is not differentiable in the interval \((-3, 3)\) are:
1. \( x = -2 \)
2. \( x = -1 \)
3. \( x = 1 \)
4. \( x = 2 \)

Thus, there are a total of **4 points** where \( f(x) \) is not differentiable.

### Final Answer
The number of points where \( f \) is not differentiable in \((-3, 3)\) is **4**.

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