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

Question

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?

Answer 1

Let consider for example $f(x)=x^4$ at the origin

and compare with $f(x)=x^3$ at the origin.