1. Vertical tangent or cusp

Hi.

I am having difficulty understanding the difference between a vertical tangent or a cusp, and when they each occur. Can someone please describe to me the difference between the two and when you know they happen? Provide examples please. Thanks!

2. Originally Posted by skeske1234
Hi.

I am having difficulty understanding the difference between a vertical tangent or a cusp, and when they each occur. Can someone please describe to me the difference between the two and when you know they happen? Provide examples please. Thanks!
consider two curves ...

$\textcolor{green}{y = \sqrt[3]{x}}$ has a vertical tangent and an inflection point at x = 0

$\textcolor{red}{y = |\sqrt[3]{x}|}$ has a cusp (a vertical tangent from both sides of x = 0), but no inflection point at x = 0

3. why doesn't the cusp have an inflection point at x=0 again?

4. Originally Posted by skeske1234
why doesn't the cusp have an inflection point at x=0 again?
because the concavity does not change ... the function is concave down on both sides of x = 0.