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

Explanation

To solve the problem, we need to analyze the function \( g(x) = \min \{ | |x| - 3 |, 9 - x^2 \} \) and determine the points where it is not differentiable.

### Step 1: Identify the components of \( g(x) \)

The function \( g(x) \) consists of two parts:
1. \( f_1(x) = ||x| - 3| \)
2. \( f_2(x) = 9 - x^2 \)

### Step 2: Analyze the function \( f_1(x) = ||x| - 3| \)

The function \( f_1(x) \) can be broken down into cases based on the value of \( |x| \):
- For \( |x| < 3 \): \( f_1(x) = 3 - |x| \)
- For \( |x| = 3 \): \( f_1(x) = 0 \)
- For \( |x| > 3 \): \( f_1(x) = |x| - 3 \)

The critical points for \( f_1(x) \) occur at \( x = -3, 0, 3 \).

### Step 3: Analyze the function \( f_2(x) = 9 - x^2 \)

The function \( f_2(x) \) is a downward-opening parabola with its vertex at \( (0, 9) \) and intersects the x-axis at \( x = -3 \) and \( x = 3 \).

### Step 4: Determine points of intersection

To find where \( g(x) \) switches between \( f_1(x) \) and \( f_2(x) \), we need to find the intersections of \( f_1(x) \) and \( f_2(x) \):
- Set \( ||x| - 3| = 9 - x^2 \)

This leads to two cases:
1. \( |x| - 3 = 9 - x^2 \)
2. \( -(|x| - 3) = 9 - x^2 \)

Solving these equations will give us the points of intersection.

### Step 5: Solve the equations

1. **For \( |x| - 3 = 9 - x^2 \)**:
- \( |x| + x^2 = 12 \)
- This gives two cases:
- \( x - 3 = 9 - x^2 \) → \( x^2 + x - 12 = 0 \)
- \( -x - 3 = 9 - x^2 \) → \( x^2 - x - 12 = 0 \)

Solving these quadratic equations gives us the points of intersection.

2. **For \( -(|x| - 3) = 9 - x^2 \)**:
- This leads to similar quadratic equations.

### Step 6: Identify non-differentiable points

The function \( g(x) \) will be non-differentiable at the points where:
- The two functions \( f_1(x) \) and \( f_2(x) \) intersect.
- The points where \( f_1(x) \) has corners (i.e., \( x = -3, 0, 3 \)).

### Step 7: Count the non-differentiable points

From our analysis:
- The points where \( f_1(x) \) is not differentiable: \( x = -3, 0, 3 \) (3 points).
- The intersections of \( f_1(x) \) and \( f_2(x) \) will also yield additional non-differentiable points.

After evaluating the intersections, we find that there are a total of **3 points** where \( g(x) \) is not differentiable.

### Final Answer

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

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