If `f(x)=root(3)((x^4)/|x|)` ,x `ne` 0 and f(0)=0 is:

Options

A. continuous for all x but not differentiable for any x
B. continuous and differentiable for all x
C. continuous for all x and differentiable for all `x ne 0`
D. continuous and differentiable for all `x ne 0`

Answer

To determine the continuity and differentiability of the function \( f(x) = \sqrt[3]{\frac{x^4}{|x|}} \) for \( x \neq 0 \) and \( f(0) = 0 \), we will follow these steps:

### Step 1: Rewrite the Function
The function can be rewritten based on the sign of \( x \):
- For \( x > 0 \):
\[ f(x) = \sqrt[3]{\frac{x^4}{x}} = \sqrt[3]{x^3} = x \]
- For \( x < 0 \):
\[ f(x) = \sqrt[3]{\frac{x^4}{-x}} = \sqrt[3]{-x^3} = -x \]

Thus, we can express the function as:
\[ f(x) = \begin{cases} x & \text{if } x > 0 \\ -x & \text{if } x < 0 \\ 0 & \text{if } x = 0 \end{cases} \]

### Step 2: Check Continuity at \( x = 0 \)
To check for continuity at \( x = 0 \), we need to verify:
\[ \lim_{x \to 0} f(x) = f(0) \]

Calculating the left-hand limit:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x = 0 \]

Calculating the right-hand limit:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \]

Since both limits equal \( f(0) = 0 \), we conclude that:
\[ \lim_{x \to 0} f(x) = f(0) \]
Thus, \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check Differentiability at \( x = 0 \)
To check for differentiability at \( x = 0 \), we need to compute the left-hand and right-hand derivatives.

**Left-hand derivative**:
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(h) - f(0)}{h - 0} = \lim_{h \to 0^-} \frac{-h - 0}{h} = \lim_{h \to 0^-} -1 = -1 \]

**Right-hand derivative**:
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(h) - f(0)}{h - 0} = \lim_{h \to 0^+} \frac{h - 0}{h} = \lim_{h \to 0^+} 1 = 1 \]

Since the left-hand derivative \( f'(0^-) = -1 \) and the right-hand derivative \( f'(0^+) = 1 \) are not equal, \( f(x) \) is not differentiable at \( x = 0 \).

### Conclusion
The function \( f(x) \) is continuous everywhere but not differentiable at \( x = 0 \). Therefore, the final answer is:
- Continuous for all \( x \)
- Differentiable for all \( x \neq 0 \)