Let `f(x){{:(max.{absx,x^2}","" "absxle2),(" "8-2absx","" "2ltabsxle4):}`.Let S be the set of points in the intercal (-4,4) at which f is not differentible. Then S

Options

A. is an empty set
B. equals {-2,-1,1,2}
C. equals {-2,-1,0,1,2}
D. equals {-2,2}

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined piecewise in the intervals specified. The function is given as:

\[ f(x) = \begin{cases} \max(|x|, x^2) & \text{for } |x| \leq 2 \\ 8 - 2|x| & \text{for } 2 < |x| \leq 4 \end{cases} \]

### Step 1: Identify the intervals and expressions for \( f(x) \)

1. For \( |x| \leq 2 \):
- We need to find \( \max(|x|, x^2) \).
- The function \( |x| \) is a V-shaped graph, and \( x^2 \) is a U-shaped graph.

2. For \( 2 < |x| \leq 4 \):
- The function is given as \( f(x) = 8 - 2|x| \).

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

- The critical points where the two functions \( |x| \) and \( x^2 \) intersect can be found by solving:
\[ |x| = x^2 \]
This gives us two equations:
- \( x = x^2 \) (i.e., \( x(x-1) = 0 \) giving \( x = 0, 1 \))
- \( -x = x^2 \) (i.e., \( x^2 + x = 0 \) giving \( x = 0, -1 \))

- Thus, the critical points in the interval \( |x| \leq 2 \) are \( -1, 0, 1 \).

### Step 3: Analyze the second case \( 2 < |x| \leq 4 \)

- The function \( f(x) = 8 - 2|x| \) is linear and continuous, but we need to check the points where it transitions from the first case to the second case, which are \( x = -2 \) and \( x = 2 \).

### Step 4: Identify points of non-differentiability

- The function \( f(x) \) can be non-differentiable at points where:
- The expressions change (i.e., at the boundaries of the intervals).
- The function has sharp turns.

From our analysis, the points where \( f(x) \) is not differentiable are:
- \( x = -2 \) (transition point)
- \( x = -1 \) (sharp turn)
- \( x = 0 \) (sharp turn)
- \( x = 1 \) (sharp turn)
- \( x = 2 \) (transition point)

### Step 5: Compile the set \( S \)

Thus, the set \( S \) of points in the interval \( (-4, 4) \) where \( f(x) \) is not differentiable is:
\[ S = \{-2, -1, 0, 1, 2\} \]

### Final Answer

The final answer is:
\[ S = \{-2, -1, 0, 1, 2\} \]