If f(x) = 0 for `x lt 0 and f(x)` is differentiable at x = 0, then for `x gt 0, f(x)` may be

Options

A. A. `x^(2)`
B. B. x
C. C. `-x`
D. D. `-x^(3//2)`

Answer

To solve the problem, we need to analyze the function \( f(x) \) given the conditions.

### Step-by-Step Solution:

1. **Understanding the Function**: We know that \( f(x) = 0 \) for \( x < 0 \). This means that the function is equal to zero for all negative values of \( x \).

2. **Differentiability at \( x = 0 \)**: The function \( f(x) \) is said to be differentiable at \( x = 0 \). For a function to be differentiable at a point, it must be continuous at that point, and the left-hand derivative (LHD) must equal the right-hand derivative (RHD).

3. **Finding Left-Hand Limit**: Since \( f(x) = 0 \) for \( x < 0 \), we have:
\[ \lim_{x \to 0^-} f(x) = 0 \]

4. **Finding Right-Hand Limit**: For \( x > 0 \), we need to check the options provided to see if they can maintain the differentiability condition at \( x = 0 \).

5. **Checking Each Option**:
- **Option A: \( f(x) = x^2 \)** (for \( x > 0 \)):
- \( \lim_{x \to 0^+} f(x) = 0^2 = 0 \)
- LHD = RHD at \( x = 0 \) since both are equal to 0. This option is valid.

- **Option B: \( f(x) = x \)** (for \( x > 0 \)):
- \( \lim_{x \to 0^+} f(x) = 0 \)
- LHD = 0 (from the left) and RHD = 1 (from the right). This option is not valid.

- **Option C: \( f(x) = -x \)** (for \( x > 0 \)):
- \( \lim_{x \to 0^+} f(x) = 0 \)
- LHD = 0 and RHD = -1. This option is not valid.

- **Option D: \( f(x) = -x^{3/2} \)** (for \( x > 0 \)):
- \( \lim_{x \to 0^+} f(x) = 0 \)
- LHD = 0 and RHD = 0. This option is valid.

6. **Conclusion**: The options that maintain differentiability at \( x = 0 \) are \( f(x) = x^2 \) and \( f(x) = -x^{3/2} \). However, since the question asks for what \( f(x) \) may be, we can conclude that \( f(x) \) may be \( x^2 \).

### Final Answer:
The function \( f(x) \) may be \( x^2 \) for \( x > 0 \).