A necessary but not sufficient condition for a point of inflection is that $$f''(x)=0$$ If the second derivative is 0 and the point is not a point of inflection, Wikipedia tells me that is called an undulation point, which apparently means
a point on a curve where the curvature vanishes but does not change sign.
An example given is $f(x)=x^4$ at $(0,0)$.
What I do not understand is why $(0,0)$ is not classified as a local minimum? Surely for $f(x)=x^4$ the first derivative changes polarity either side of $(0,0)$.
Is there a better example of an 'undulation point' for a smooth function? Do 'undulation points' exist for more than one variable or is the condition that all partial derivatives are zero a necessary and sufficient one?