If `f (x)=|||x|-2|+p|` have more than 3 points non-derivability then the value of p can be :

Options

A. 0
B. `-1`
C. `-2`
D. 2

Answer

To solve the problem, we need to analyze the function \( f(x) = |||x| - 2| + p| \) and determine the values of \( p \) for which the function has more than three points of non-derivability.

### Step 1: Identify the points of non-derivability of the function

The function \( f(x) \) is composed of nested absolute values. The points of non-derivability occur where the expression inside the absolute values equals zero.

1. **Innermost Absolute Value**:
- The expression \( |x| - 2 = 0 \) gives us \( |x| = 2 \), which results in \( x = 2 \) and \( x = -2 \).

2. **Middle Absolute Value**:
- The expression \( ||x| - 2| + p = 0 \) will also give us points of non-derivability. We need to analyze this expression further.
- The equation \( ||x| - 2| + p = 0 \) implies \( ||x| - 2| = -p \). Since the absolute value cannot be negative, we need \( p \leq 0 \).

### Step 2: Determine the critical points based on \( p \)

Now, we will analyze the critical points based on different values of \( p \):

1. **If \( p = 0 \)**:
- The equation simplifies to \( ||x| - 2| = 0 \), which gives us two points: \( x = 2 \) and \( x = -2 \).

2. **If \( p < 0 \)**:
- The equation \( ||x| - 2| = -p \) will yield additional points.
- For \( p = -1 \), we have \( ||x| - 2| = 1 \), leading to four points: \( x = 1, -1, 3, -3 \).
- For \( p = -2 \), we have \( ||x| - 2| = 2 \), leading to four points: \( x = 0, 4, -4 \).

### Step 3: Count the total points of non-derivability

To have more than three points of non-derivability, we need to consider the cases where \( p < 0 \):

- For \( p = -1 \): Points of non-derivability are \( -3, -2, -1, 1, 2, 3 \) (total of 6 points).
- For \( p = -2 \): Points of non-derivability are \( -4, -2, 0, 2, 4 \) (total of 5 points).

### Conclusion

Thus, the values of \( p \) for which \( f(x) \) has more than three points of non-derivability are \( p < 0 \). Specifically, \( p \) can be any value less than or equal to -1.

### Final Answer:
The values of \( p \) can be \( p = -1 \) or \( p = -2 \).

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