Let `f(x)={{:("max"{|x|","x^(2)},|x|le3),(12-|x|,3lt|x|le12):}`. If S is the set of points in the interval `(-12, 12)` at which f is not differentiable, then S is

Options

A. equal to `{-3, 3}`
B. equal to `{-3, -1, 1, 3}`
C. an empty set
D. equal to `{-3, -1, 0, 1, 3}`

Answer

To determine the set of points \( S \) in the interval \( (-12, 12) \) where the function \( f(x) \) is not differentiable, we will analyze the function step by step.

### Step 1: Understand the function definition
The function \( f(x) \) is defined piecewise as follows:
- For \( |x| \leq 3 \): \( f(x) = \max(|x|, x^2) \)
- For \( 3 < |x| \leq 12 \): \( f(x) = 12 - |x| \)

### Step 2: Identify critical points
To find where \( f(x) \) is not differentiable, we need to identify points where the function changes its definition or has sharp corners. These points typically occur at the boundaries of the defined intervals and at points where the maximum function changes.

1. **For \( |x| \leq 3 \)**:
- The critical points to consider are \( x = -3, -1, 0, 1, 3 \).

2. **For \( 3 < |x| \leq 12 \)**:
- The function changes at \( x = 3 \) and \( x = -3 \).

### Step 3: Analyze the intervals
- **Interval \( (-12, -3) \)**: Here, \( f(x) = 12 + x \) (a linear function).
- **Interval \( (-3, -1) \)**: Here, \( f(x) = x^2 \).
- **Interval \( (-1, 0) \)**: Here, \( f(x) = |x| = -x \).
- **Interval \( (0, 1) \)**: Here, \( f(x) = |x| = x \).
- **Interval \( (1, 3) \)**: Here, \( f(x) = x^2 \).
- **Interval \( (3, 12) \)**: Here, \( f(x) = 12 - x \).

### Step 4: Check differentiability at critical points
1. **At \( x = -3 \)**:
- Left-hand derivative: \( f'(x) = -1 \) (from \( 12 + x \)).
- Right-hand derivative: \( f'(x) = -6 \) (from \( x^2 \)).
- Not differentiable.

2. **At \( x = -1 \)**:
- Left-hand derivative: \( f'(x) = -2 \) (from \( x^2 \)).
- Right-hand derivative: \( f'(x) = 1 \) (from \( -x \)).
- Not differentiable.

3. **At \( x = 0 \)**:
- Left-hand derivative: \( f'(x) = -1 \) (from \( -x \)).
- Right-hand derivative: \( f'(x) = 1 \) (from \( x \)).
- Not differentiable.

4. **At \( x = 1 \)**:
- Left-hand derivative: \( f'(x) = 1 \) (from \( x \)).
- Right-hand derivative: \( f'(x) = 2 \) (from \( x^2 \)).
- Not differentiable.

5. **At \( x = 3 \)**:
- Left-hand derivative: \( f'(x) = 6 \) (from \( x^2 \)).
- Right-hand derivative: \( f'(x) = -1 \) (from \( 12 - x \)).
- Not differentiable.

### Step 5: Compile the set of non-differentiable points
The points where \( f(x) \) is not differentiable are:
- \( x = -3 \)
- \( x = -1 \)
- \( x = 0 \)
- \( x = 1 \)
- \( x = 3 \)

Thus, the set \( S \) is:
\[ S = \{-3, -1, 0, 1, 3\} \]

### Final Answer
The set \( S \) of points in the interval \( (-12, 12) \) at which \( f \) is not differentiable is:
\[ S = \{-3, -1, 0, 1, 3\} \]