Question

(d) Prove that the function \( f(x) = |x| \) is not differentiable at \( x = 0 \):

Hint:

Check both left-hand and right-hand derivatives to determine differentiability at a point.

Answer 1

The function \( f(x) = |x| \) can be written as: \[ f(x) = \begin{cases} x & \text{if } x \geq 0,
-x & \text{if } x<0. \end{cases} \] The left-hand derivative at \( x = 0 \): \[ f'(0^-) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-h - 0}{h} = -1. \] The right-hand derivative at \( x = 0 \): \[ f'(0^+) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h - 0}{h} = 1. \] Since \( f'(0^-) \neq f'(0^+) \), \( f(x) \) is not differentiable at \( x = 0 \).