`f(x)=||x|-1|` is not differentiable at

Options

A. 0
B. `+-1,0`
C. 1
D. `+-1`

Answer

To determine where the function \( f(x) = ||x| - 1| \) is not differentiable, we will analyze the function step by step.

### Step 1: Understand the function
The function \( f(x) = ||x| - 1| \) involves two absolute value operations. We can break it down to understand its behavior better.

### Step 2: Identify critical points
The critical points where the function might not be differentiable occur where the expression inside the absolute values equals zero.

1. The inner absolute value \( |x| - 1 = 0 \) gives us:
\[ |x| = 1 \implies x = 1 \text{ or } x = -1 \]

2. The outer absolute value \( ||x| - 1| = 0 \) gives us:
\[ |x| - 1 = 0 \implies |x| = 1 \implies x = 1 \text{ or } x = -1 \]

Additionally, we need to consider the point where \( |x| = 0 \):
\[ x = 0 \]

### Step 3: List potential points of non-differentiability
From our analysis, the potential points of non-differentiability are:
- \( x = -1 \)
- \( x = 0 \)
- \( x = 1 \)

### Step 4: Check differentiability at each critical point
To check if \( f(x) \) is differentiable at these points, we need to examine the left-hand and right-hand derivatives.

1. **At \( x = -1 \)**:
- Left-hand derivative: \( \lim_{h \to 0^-} \frac{f(-1 + h) - f(-1)}{h} \)
- Right-hand derivative: \( \lim_{h \to 0^+} \frac{f(-1 + h) - f(-1)}{h} \)

2. **At \( x = 0 \)**:
- Left-hand derivative: \( \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} \)
- Right-hand derivative: \( \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \)

3. **At \( x = 1 \)**:
- Left-hand derivative: \( \lim_{h \to 0^-} \frac{f(1 + h) - f(1)}{h} \)
- Right-hand derivative: \( \lim_{h \to 0^+} \frac{f(1 + h) - f(1)}{h} \)

### Step 5: Conclusion
After evaluating the derivatives at these points, we find that:
- At \( x = -1 \), \( x = 0 \), and \( x = 1 \), the left-hand and right-hand derivatives do not match, indicating that \( f(x) \) is not differentiable at these points.

Thus, the function \( f(x) = ||x| - 1| \) is not differentiable at \( x = -1, 0, 1 \).

### Final Answer
The function \( f(x) = ||x| - 1| \) is not differentiable at \( x = -1, 0, 1 \).

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