Would f ''(a)=0 and f ''(x) does not change sign at x=a result in an inflection point?

My note says: " If f ''(a)=0 and f ''(x) changes sign at x=a, there is a point of inflection at (a, f(a))."

I was wondering how the original graph would appear if f ''(a)=0 and f ''(x) does not change sign at x=a, would there still be a point of inflection?

• If $f''$ does not change sign, no inflection point occurs. Just like how, if $f'(a)=0$ but $f'$ doesn't change sign at $x=a$, $f$ does not have a relative extremum at $x=a$. – Franklin Pezzuti Dyer Mar 19 '18 at 23:26
• My note also says that a point of inflection occurs where f''(x) is undefined but how is this possible? – Timmy Mar 19 '18 at 23:44
• Here's how to ask a good question. In particular, note that we use MathJax here. – Shaun Mar 20 '18 at 0:28

Let consider for example $f(x)=x^4$ at the origin
and compare with $f(x)=x^3$ at the origin.
• @Timmy Yes indeed it is a simple example of what occurs if $f''(0)=0$ but $f''$ doesn't change sign at $x=0$. Maybe you are looking for something else? – user Mar 19 '18 at 23:43
• @Timmy Yes think to $x^\frac13$ i.stack.imgur.com/Ljk9m.png the slope at 0 is vertical and it is an inflection point thus f'(x) and f''(x) are undefined. – user Mar 19 '18 at 23:49