Graph with corner, cusp and vertical tangent

Problem

Sketch the graph of a function that has: a corner point at $-1$, a cusp at $0$, an inflection point with vertical tangent at $2$.

Solution

It is enough to “glue together” three local behaviours already known, each translated to the required point. An explicit example is: $f(x) = |x+1| \;+\; \sqrt{|x|} \;+\; \sqrt[3]{x-2}.$

• At $x=-1$ the term $|x+1|$ creates an edge with half-tangents of slope $-1$ and $+1$ (the other terms are smooth there): corner point.
• At $x=0$ the term $\sqrt{|x|}$ has one-sided derivatives $-\infty$ and $+\infty$: cusp.
• At $x=2$ the term $\sqrt[3]{x-2}$ has $f'\to+\infty$ on both sides: inflection point with vertical tangent.

In the drawing: a “V”-shaped edge at $x=-1$, a downward-pointing spike at $x=0$, and a rise with vertical tangent (vertical “S”-shaped curve) at $x=2$.

Links

Topics: Derivatives
Concepts: Cusp · Corner point · Non-differentiable point · Vertical tangent
Skills: Reason by cases · Sketch a graph
Exercise type: Reading a graph