Horizontal point of inflexion? (1 Viewer)

[green_appletini]

Hey guys,

i was just wondering, what's the proof for a horizontal point of inflexion?
I have always said it's a horizontal point of inflexion when:

- 1st derivative= 0
- 2nd derivative= 0
- test for change in concavity and if there is a change, it means it's a horizontal pt of inflexion

but a friend said to me that you don't need to test for change in concavity and that u only do so for other points of inflexion. so can someone help? are the 1st 2 steps i listed above sufficient to say that a point is a horizontal point of inflexion?

thanks!
-marilia- xoxo

 

[Tommy_Lamp]

f'(x) = 0, since gradient is zero
and yes you would have to test for concavity because it could just be a turning point.

 

[Xayma]

Yes you do.

y=x⁴ does not have a horizontal point of inflexion however at x=0
dy/dx=0
d²y/dx²=0