The number of points where g(x) = min { ||x| -3|, ` 9 -x^(2)} ( x in R )` is not differentiable is _______

Answer

To find the number of points where the function \( g(x) = \min \{ ||x| - 3|, 9 - x^2 \} \) is not differentiable, we will analyze the two functions involved in the minimum operation: \( ||x| - 3| \) and \( 9 - x^2 \).

### Step 1: Analyze the function \( ||x| - 3| \)

1. **Find the points where \( |x| - 3 = 0 \)**:
- This occurs when \( |x| = 3 \), which gives \( x = 3 \) and \( x = -3 \).

2. **Determine the behavior of \( ||x| - 3| \)**:
- For \( x < -3 \): \( ||x| - 3| = -|x| + 3 = -x - 3 \)
- For \( -3 \leq x \leq 3 \): \( ||x| - 3| = |x| - 3 \)
- For \( x > 3 \): \( ||x| - 3| = |x| - 3 = x - 3 \)

3. **Check for non-differentiability**:
- The function \( ||x| - 3| \) is not differentiable at \( x = -3 \) and \( x = 3 \) (points where the absolute value function changes).

### Step 2: Analyze the function \( 9 - x^2 \)

1. **Determine the vertex and intercepts**:
- The function \( 9 - x^2 \) is a downward-opening parabola with vertex at \( (0, 9) \) and intercepts at \( x = -3 \) and \( x = 3 \).

2. **Check for differentiability**:
- The function \( 9 - x^2 \) is a polynomial and is differentiable everywhere.

### Step 3: Combine the two functions

1. **Identify the points where the minimum changes**:
- The minimum function \( g(x) \) will change at the points where \( ||x| - 3| = 9 - x^2 \).
- We need to check the intersections of the two functions:
- Set \( ||x| - 3| = 9 - x^2 \) and solve for \( x \).

2. **Points of intersection**:
- For \( x < -3 \): \( -x - 3 = 9 - x^2 \) leads to \( x^2 - x - 12 = 0 \) which factors to \( (x - 4)(x + 3) = 0 \) giving \( x = 4 \) (not valid) and \( x = -3 \).
- For \( -3 \leq x \leq 3 \): \( |x| - 3 = 9 - x^2 \) leads to \( x^2 + |x| - 12 = 0 \) which gives \( x = -3, 3 \).
- For \( x > 3 \): \( x - 3 = 9 - x^2 \) leads to \( x^2 + x - 12 = 0 \) which factors to \( (x - 3)(x + 4) = 0 \) giving \( x = 3 \) (valid) and \( x = -4 \) (not valid).

### Step 4: Count the points of non-differentiability

1. **List the points**:
- From \( ||x| - 3| \): \( x = -3, 3 \)
- From the intersection of both functions: \( x = -3, 3 \)

2. **Conclusion**:
- The function \( g(x) \) is not differentiable at \( x = -3 \) and \( x = 3 \) due to the changes in the definition of the minimum function.
- Additionally, the function is not differentiable at \( x = 0 \) (the vertex of the parabola).

Thus, the total number of points where \( g(x) \) is not differentiable is **3**.

### Final Answer:
The number of points where \( g(x) \) is not differentiable is **3**.

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