The set of all points where the function `f(x)=3sqrt(x^2|x|)` is differentiable, is

Options

A. `[0,oo)`
B. `(0,oo)`
C. `(-oo,oo)`
D. `(-oo0) uu (0,oo)`

Explanation

To determine the set of all points where the function \( f(x) = 3\sqrt{x^2 |x|} \) is differentiable, we can follow these steps:

### Step 1: Rewrite the function
The function can be rewritten using the definition of the absolute value. We know that:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

Thus, we can express \( f(x) \) as:
\[ f(x) = 3\sqrt{x^2 |x|} = 3\sqrt{x^2 \cdot x} = 3\sqrt{x^3} \quad \text{for } x \geq 0 \]
\[ f(x) = 3\sqrt{x^2 \cdot (-x)} = 3\sqrt{-x^3} \quad \text{for } x < 0 \]

### Step 2: Simplify the function
For \( x \geq 0 \):
\[ f(x) = 3x^{3/2} \]
For \( x < 0 \):
\[ f(x) = 3(-x)^{3/2} = -3(-x)^{3/2} \]

### Step 3: Analyze differentiability
Now we need to check the differentiability of \( f(x) \) at \( x = 0 \) and for \( x \neq 0 \).

1. **For \( x > 0 \)**:
- The function \( f(x) = 3x^{3/2} \) is differentiable since it is a polynomial function.

2. **For \( x < 0 \)**:
- The function \( f(x) = -3(-x)^{3/2} \) is also differentiable for \( x < 0 \) since it is a composition of differentiable functions.

3. **At \( x = 0 \)**:
- We need to check the left-hand and right-hand derivatives.
- Right-hand derivative at \( x = 0 \):
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{3h^{3/2}}{h} = \lim_{h \to 0^+} 3h^{1/2} = 0 \]
- Left-hand derivative at \( x = 0 \):
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-3(-h)^{3/2}}{h} = \lim_{h \to 0^-} -3(-h)^{1/2} = 0 \]

Since both the left-hand and right-hand derivatives at \( x = 0 \) are equal, \( f(x) \) is differentiable at \( x = 0 \).

### Conclusion
The function \( f(x) = 3\sqrt{x^2 |x|} \) is differentiable for all \( x \in \mathbb{R} \).

### Final Answer
The set of all points where the function is differentiable is \( (-\infty, \infty) \).