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 determine if the function $f(x) = |x|$ is differentiable at $x = 0$, we must assess the existence of its derivative at that point. 1. The function $f(x) = |x|$ is defined piecewise: \[ f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x<0 \end{cases} \] 2. For $x>0$, the derivative of $f(x)$ is $f'(x) = 1$. For $x<0$, the derivative of $f(x)$ is $f'(x) = -1$. 3. At $x = 0$, we evaluate the left-hand and right-hand derivatives: * 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 at $x = 0$ are unequal, the derivative does not exist at this point. Consequently, the function $f(x) = |x|$ is not differentiable at $x = 0$.