Let f (x) = 10 - |x-5| , `x in R,` then the set of all values of x at which f (f(x)) is not differentiable is

Options

A. {0,5,10}
B. {5,10}
C. {0,5,10,15}
D. {5,10,15}

Answer

To solve the problem, we need to determine the set of all values of \( x \) at which \( f(f(x)) \) is not differentiable, where \( f(x) = 10 - |x - 5| \).

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) = 10 - |x - 5| \) can be rewritten based on the definition of absolute value:

- For \( x < 5 \):
\[ f(x) = 10 - (5 - x) = x + 5 \]

- For \( x = 5 \):
\[ f(5) = 10 - |5 - 5| = 10 \]

- For \( x > 5 \):
\[ f(x) = 10 - (x - 5) = 15 - x \]

Thus, we can summarize \( f(x) \) as:
\[ f(x) = \begin{cases} x + 5 & \text{if } x < 5 \\ 10 & \text{if } x = 5 \\ 15 - x & \text{if } x > 5 \end{cases} \]

### Step 2: Find \( f(f(x)) \)

Now we need to find \( f(f(x)) \). We will consider the three cases for \( f(x) \):

1. **Case 1: \( x < 5 \)**
- Here, \( f(x) = x + 5 \).
- Since \( x + 5 \geq 5 \) for \( x < 5 \), we have:
\[ f(f(x)) = f(x + 5) = 15 - (x + 5) = 10 - x \]

2. **Case 2: \( x = 5 \)**
- Here, \( f(5) = 10 \).
- Thus:
\[ f(f(5)) = f(10) = 15 - 10 = 5 \]

3. **Case 3: \( x > 5 \)**
- Here, \( f(x) = 15 - x \).
- Since \( 15 - x \leq 10 \) for \( x \geq 5 \), we consider two subcases:
- If \( 5 < x < 10 \):
\[ f(f(x)) = f(15 - x) = 10 - (15 - x) = x - 5 \]
- If \( x = 10 \):
\[ f(f(10)) = f(5) = 10 \]
- If \( x > 10 \):
\[ f(f(x)) = f(15 - x) = 10 - (x - 15) = 25 - x \]

### Step 3: Determine where \( f(f(x)) \) is not differentiable

The function \( f(f(x)) \) is given by:
\[ f(f(x)) = \begin{cases} 10 - x & \text{if } x < 5 \\ 5 & \text{if } x = 5 \\ x - 5 & \text{if } 5 < x < 10 \\ 10 & \text{if } x = 10 \\ 25 - x & \text{if } x > 10 \end{cases} \]

Now we check the points where \( f(f(x)) \) changes its definition:
- At \( x = 5 \): The left-hand limit is \( 10 - 5 = 5 \) and the right-hand limit is \( 5 \). The function is continuous but not differentiable since the derivative changes from \(-1\) to \(0\).
- At \( x = 10 \): The left-hand limit is \( 10 - 10 = 0 \) and the right-hand limit is \( 25 - 10 = 15 \). The function is continuous but not differentiable since the derivative changes from \(1\) to \(-1\).

### Conclusion

The points where \( f(f(x)) \) is not differentiable are \( x = 0, 5, 10 \). Thus, the set of all values of \( x \) at which \( f(f(x)) \) is not differentiable is:

\[ \{0, 5, 10\} \]