Question

Check the differentiability of the function $f(x) = |x|$ at $x = 0$.

Hint:

A function is differentiable at a point if the left-hand and right-hand derivatives at that point are equal.

Answer 1

To check the differentiability of the function $f(x) = |x|$ at $x = 0$, we need to check if the derivative exists at that point. 1. The function $f(x) = |x|$ is defined as \[ f(x) = \begin{cases} x, & \text{if } x \geq 0
-x, & \text{if } x<0 \end{cases} \] 2. The derivative of $f(x)$ for $x>0$ is \[ f'(x) = 1. \] For $x<0$, the derivative is \[ f'(x) = -1. \] 3. At $x = 0$, we need to check if the left-hand derivative and the right-hand derivative exist and are equal: - Left-hand derivative: \[ \lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0^-} \frac{-x - 0}{x} = -1. \] - Right-hand derivative: \[ \lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0^+} \frac{x - 0}{x} = 1. \] Since the left-hand and right-hand derivatives are not equal, the derivative does not exist at $x = 0$. Therefore, the function $f(x) = |x|$ is not differentiable at $x = 0$.