# Vertical tangent or cusp

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• Aug 20th 2009, 01:32 PM
skeske1234
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!
• Aug 20th 2009, 03:02 PM
skeeter
Quote:

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
• Aug 20th 2009, 04:47 PM
skeske1234
why doesn't the cusp have an inflection point at x=0 again?
• Aug 20th 2009, 05:01 PM
skeeter
Quote:

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.