# Cusps and Points of Inflection

Can cusps be considered points of inflection?

I'm getting conflicting information but my thought process is that cusps cannot be points of inflection?

Can points of inflection exist when there is a vertical tangent to the graph? Assume there is change in concavity and the function is continuous.

• Is there a derivative at a cusp? – Henry Nov 10 '15 at 23:55
• derivative doesnt exist at a cusp – user264985 Nov 10 '15 at 23:55
• So it may depend on whether smoothness is part of your definition of a point of inflection or whether you use a definition that does not require smoothness (e.g. any change between convexity and concavity). My personal inclination would be to say not. – Henry Nov 11 '15 at 0:03

no. look at the graph of $y = x^{2/3}.$ this has a cusp at $(0,0)$ but concave down on $(-\infty, \infty)$ and $(0,0)$ is certainly not a point of inflection.