Classifying points of non-differentiability

Statement

Classify the point $x_0 = 0$ for each function: (a) $f(x) = \sqrt[3]{x^2}$; (b) $f(x) = x^{2/3}\sin x$; (c) $f(x) = |x^2-1|$ at $x_0 = 1$; (d) $f(x) = \sqrt[5]{x}$.

Solution

(a) $f(x)=x^{2/3}$, $f'(x)=\frac{2}{3}x^{-1/3}=\dfrac{2}{3\sqrt[3]{x}}$. As $x\to 0^+$: $f'\to+\infty$; as $x\to 0^-$: $f'\to-\infty$. Infinite with opposite signs: cusp.

(b) Difference quotient at $0$: $\dfrac{x^{2/3}\sin x}{x} = x^{-1/3}\sin x = x^{2/3}\cdot\dfrac{\sin x}{x}\to 0$. The derivative exists, $f'(0)=0$: a differentiable point (stationary), not a point of non-differentiability.

(c) Near $x_0=1$: for $x>1$, $x^2-1>0$, $f=x^2-1$, $f'_+(1)=2$; for $x<1$, $f=1-x^2$, $f'_-(1)=-2$. Finite and different: corner point.

(d) $f(x)=x^{1/5}$, $f'(x)=\frac{1}{5}x^{-4/5}\to+\infty$ from both sides (same sign): inflection with vertical tangent.

Links

Topics: Derivatives
Concepts: Cusp · Corner point · Point of non-differentiability · Vertical tangent
Methods: Classifying points of non-differentiability
Skills: Computing limits · Reasoning by cases
Exercise type: Derivative computation