Let ` f(x) = 15 -|x-10|, x in R`. Then, the set of all values of x, at which the function, ` g(x) = f(f(x))` is not differentiable, is

Options

A. {10}
B. {5, 10, 15}
C. {5, 10, 15, 20}
D. {10, 15}

Explanation

To solve the problem, we need to find the set of all values of \( x \) at which the function \( g(x) = f(f(x)) \) is not differentiable, where \( f(x) = 15 - |x - 10| \).

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

The function is given by:

\[ f(x) = 15 - |x - 10| \]

This function has a vertex at \( x = 10 \). We can analyze the behavior of \( f(x) \) based on the definition of the absolute value.

### Step 2: Analyze \( f(x) \) for different intervals

1. **For \( x < 10 \)**:
\[ f(x) = 15 - (10 - x) = x + 5 \]

2. **For \( x \geq 10 \)**:
\[ f(x) = 15 - (x - 10) = 25 - x \]

### Step 3: Find \( f(f(x)) \)

Now we need to find \( g(x) = f(f(x)) \).

#### Case 1: \( x < 10 \)

In this case, \( f(x) = x + 5 \). We need to check if \( x + 5 \) is less than or greater than 10.

- If \( x + 5 < 10 \) (which is true for \( x < 5 \)):
\[ f(f(x)) = f(x + 5) = (x + 5) + 5 = x + 10 \]

- If \( x + 5 \geq 10 \) (which is true for \( x \geq 5 \)):
\[ f(f(x)) = f(x + 5) = 25 - (x + 5) = 20 - x \]

#### Case 2: \( x \geq 10 \)

In this case, \( f(x) = 25 - x \). We need to check if \( 25 - x \) is less than or greater than 10.

- If \( 25 - x < 10 \) (which is true for \( x > 15 \)):
\[ f(f(x)) = f(25 - x) = (25 - x) + 5 = 30 - x \]

- If \( 25 - x \geq 10 \) (which is true for \( x \leq 15 \)):
\[ f(f(x)) = f(25 - x) = 25 - (25 - x) = x \]

### Step 4: Combine the results

Now we have the following piecewise function for \( g(x) \):

\[ g(x) = \begin{cases} x + 10 & \text{for } x < 5 \\ 20 - x & \text{for } 5 \leq x < 10 \\ x & \text{for } 10 \leq x < 15 \\ 30 - x & \text{for } x \geq 15 \end{cases} \]

### Step 5: Identify points of non-differentiability

To find where \( g(x) \) is not differentiable, we need to check the points where the function changes its definition:

1. At \( x = 5 \): The function changes from \( 20 - x \) to \( x \).
2. At \( x = 10 \): The function changes from \( x + 10 \) to \( 20 - x \).
3. At \( x = 15 \): The function changes from \( x \) to \( 30 - x \).

### Conclusion

The set of all values of \( x \) at which the function \( g(x) \) is not differentiable is:

\[ \{5, 10, 15\} \]