If `f(x)=x^3sgn(x),` then

Options

A. f is derivable at x=0
B. f is continuous but not derivable at x=0
C. LHD at x=0 is 1
D. RHD at x=0 is 1

Answer

To solve the problem, we need to analyze the function \( f(x) = x^3 \text{sgn}(x) \) and determine its continuity and differentiability at \( x = 0 \).

### Step 1: Define the function based on the signum function
The signum function, \( \text{sgn}(x) \), is defined as follows:
- \( \text{sgn}(x) = 1 \) if \( x > 0 \)
- \( \text{sgn}(x) = 0 \) if \( x = 0 \)
- \( \text{sgn}(x) = -1 \) if \( x < 0 \)

Using this definition, we can express \( f(x) \) in piecewise form:
\[ f(x) = \begin{cases} -x^3 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ x^3 & \text{if } x > 0 \end{cases} \]

### Step 2: Check for continuity at \( x = 0 \)
To check for continuity at \( x = 0 \), we need to evaluate the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point.

1. **Left-hand limit (LHL)**:
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x^3 = -0^3 = 0 \]

2. **Right-hand limit (RHL)**:
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^3 = 0^3 = 0 \]

3. **Value of the function at \( x = 0 \)**:
\[ f(0) = 0 \]

Since LHL = RHL = \( f(0) = 0 \), we conclude that \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check for differentiability at \( x = 0 \)
To check for differentiability at \( x = 0 \), we need to find the left-hand derivative (LHD) and right-hand derivative (RHD).

1. **Left-hand derivative (LHD)**:
\[ \text{LHD} = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-h^3 - 0}{h} = \lim_{h \to 0^-} -h^2 = 0 \]

2. **Right-hand derivative (RHD)**:
\[ \text{RHD} = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^3 - 0}{h} = \lim_{h \to 0^+} h^2 = 0 \]

Since LHD = RHD = 0, we conclude that \( f(x) \) is differentiable at \( x = 0 \).

### Conclusion
The function \( f(x) = x^3 \text{sgn}(x) \) is both continuous and differentiable at \( x = 0 \).

### Final Answer
The function is differentiable at \( x = 0 \).

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