My question is this. What properties does a function need to have such that it has some c that satisfies the relationship $$ f'(c) = f''(c)=0 $$ In other words what functions would have their one or more of their critical numbers equal to their points of inflection. Any polynomial function that does not have a quadratic or linear term would satisfy the relationship, but that seems trivial.
[Edit] Trivial here means that the solution only includes terms that are easily manipulated at specific values such as using $sin(x)$ at $x = 0$. So $x^3-3x^2+3x-1$ is trivial because it is just $x^3$ shifted to the right one unit. This is a link to what I would consider non-trivial polynomial examples.
However, I am looking for an answer that either, involves any of the following: logarithms, trigonometric, other non-elementary functions. Or a polynomial example where $f'(c) = f''(c)=0$ occurs more than once.