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 find the set of all values of \( x \) at which \( f(f(x)) \) is not differentiable, we will follow these steps:

### Step 1: Define the function \( f(x) \)
The function is given as:
\[ f(x) = 10 - |x - 5| \]

### Step 2: Find \( f(f(x)) \)
To find \( f(f(x)) \), we first substitute \( f(x) \) into itself:
\[ f(f(x)) = f(10 - |x - 5|) \]

### Step 3: Substitute \( f(x) \) into the function
Now, we need to evaluate \( f(10 - |x - 5|) \):
\[ f(10 - |x - 5|) = 10 - |(10 - |x - 5|) - 5| \]
This simplifies to:
\[ = 10 - |5 - |x - 5|| \]

### Step 4: Analyze the expression \( |5 - |x - 5|| \)
The expression \( |5 - |x - 5| \) can change based on the value of \( x \). We need to find the critical points where this expression is not differentiable.

### Step 5: Identify points where \( |x - 5| \) changes
The expression \( |x - 5| \) changes at \( x = 5 \). Therefore, we need to consider the cases for \( x \):

1. **Case 1:** \( x < 5 \)
- Here, \( |x - 5| = 5 - x \)
- Thus, \( |5 - |x - 5|| = |5 - (5 - x)| = |x| \)

2. **Case 2:** \( x = 5 \)
- Here, \( |x - 5| = 0 \)
- Thus, \( |5 - |x - 5|| = |5 - 0| = 5 \)

3. **Case 3:** \( x > 5 \)
- Here, \( |x - 5| = x - 5 \)
- Thus, \( |5 - |x - 5|| = |5 - (x - 5)| = |10 - x| \)

### Step 6: Find points of non-differentiability
The function \( f(f(x)) \) will not be differentiable at the points where \( |5 - |x - 5|| \) changes, which are:
- At \( x = 5 \) (where \( |x - 5| \) changes)
- At \( x = 10 \) (where \( |10 - x| \) changes)

### Conclusion
The set of all values of \( x \) at which \( f(f(x)) \) is not differentiable is:
\[ \{ 5, 10 \} \]