Question

Check whether the function \( f(x) = x^2 |x| \) is differentiable at \( x = 0 \) or not.
Answer: \( f(x) \) is differentiable at \( x = 0 \).

Hint:

N/A

Answer 1

The function \( f(x) = x^2 |x| \) can be expressed as: \[ f(x) = \begin{cases} x^3 & \text{if } x \geq 0, \\ -x^3 & \text{if } x < 0. \end{cases} \] To assess differentiability at \( x = 0 \), we calculate the left-hand derivative (LHD) and the right-hand derivative (RHD).

1. Right-hand derivative (RHD): For \( x \geq 0 \), \( f'(x) = \frac{d}{dx}(x^3) = 3x^2 \). At \( x = 0 \), the RHD is \( f'_+(0) = 3(0)^2 = 0 \).

2. Left-hand derivative (LHD): For \( x < 0 \), \( f'(x) = \frac{d}{dx}(-x^3) = -3x^2 \). At \( x = 0 \), the LHD is \( f'_-(0) = -3(0)^2 = 0 \).

As \( f'_+(0) = f'_-(0) = 0 \), the derivative exists and is continuous at \( x = 0 \). Consequently, \( f(x) \) is differentiable at \( x = 0 \).