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.
Take for example $\,f(x) = g(x^3)\,$ at $\,c=0\,$ for any suitably differentiable $\,g\,$.
[ EDIT ] The most general description you can probably get is that $\,f\,$ is of the form $\,f = \int g\,$ where $\,g\,$ is a differentiable function that has a critical point $\,c\,$ where its value $\,g(c)=0\,$.
[ EDIT #2 ] More examples below.
$\sin(x)-x\;$ at $\;c=0$
$\log(1+x^n)\;$ at $\;c=0\;$ for $\,\forall n \ge 3$
$\log( 1 + \sin^2(x^2))\;$ at $\;c=0$
$x^6 - 9 x^5 + 33 x^4 - 63 x^3 + 66 x^2 - 36 x + 8\;$ at $\;c=1\,$ and $\,c=2$
$(x-x_1)^{n_1}(x-x_2)^{n_2} \ldots\;$ for any distinct $\,x_1, x_2, \ldots\;$ and odd $\;n_1, n_2, \ldots \ge 3\;$