# Inflection point

1. Nov 23, 2008

### fishingspree2

Hello,

I have read on this forum and on the internet that a function y(x) has an inflection point at x=c if y''(c) = 0. In other words, if we are asked to find inflection points of a function y(x), we need to solve y''(x) = 0

My question is, don't we also need to verify if the concavity changes from c- to c+? If we have a function that has a point x=c that verify y''(c)=0, but does not change concavity from c- to c+, then I don't think ((c, y(c)) would qualify as a inflection point.

Suppose the second derivative of some function is x2, if x=0 then the function is =0, but the function x2 is always positive, so going from 0- to 0+, there is no sign change, and thus no concavity change.

Can somebody please clarify this concept for me?

2. Nov 23, 2008

### tiny-tim

Hello fishingspree2!

Yes, y''(c) = 0 is a necessary condition, but not a sufficient one.

You need another test, to make sure it hasn't "changed its mind"!

Concavity will do.