The set of all points where `f(x) = root(3)(x^(2)|x|)-|x|-1` is not differentiable is

Options

A. {0}
B. (-1,0,1}
C. {0,1}
D. none of these

Answer

To determine the set of all points where the function \( f(x) = \sqrt[3]{x^2 |x|} - |x| - 1 \) is not differentiable, we will analyze the function step by step.

### Step 1: Rewrite the function
We start with the given function:
\[ f(x) = \sqrt[3]{x^2 |x|} - |x| - 1 \]
We can express \( x^2 |x| \) as \( |x|^3 \) because \( x^2 |x| = |x|^3 \) for all \( x \). Thus, we can rewrite the function as:
\[ f(x) = \sqrt[3]{|x|^3} - |x| - 1 \]

### Step 2: Simplify the function
The cube root of \( |x|^3 \) is simply \( |x| \), so we have:
\[ f(x) = |x| - |x| - 1 \]
This simplifies to:
\[ f(x) = -1 \]

### Step 3: Analyze differentiability
The function \( f(x) = -1 \) is a constant function. A constant function is differentiable everywhere in its domain. Therefore, \( f(x) \) is differentiable for all \( x \in \mathbb{R} \).

### Step 4: Conclusion
Since \( f(x) \) is differentiable for all real numbers, the set of all points where \( f(x) \) is not differentiable is the empty set.

Thus, the final answer is:
\[ \text{The set of all points where } f(x) \text{ is not differentiable is } \emptyset. \]

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