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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.

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  • $\begingroup$ Thanks that was what I was looking for. $\endgroup$ Commented Apr 19, 2018 at 21:44

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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.

either, involves any of the following: logarithms, trigonometric, other non-elementary functions.

  • $\sin(x)-x\;$ at $\;c=0$

  • $\log(1+x^n)\;$ at $\;c=0\;$ for $\,\forall n \ge 3$

  • $\log( 1 + \sin^2(x^2))\;$ at $\;c=0$

Or a polynomial example where $f'(c) = f''(c)=0$ occurs more than once.

  • $x^6 - 9 x^5 + 33 x^4 - 63 x^3 + 66 x^2 - 36 x + 8\;$ 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 $\;n_1, n_2, \ldots \ge 3\;$

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  • $\begingroup$ I was looking for a non trivial answer, and I even said "Any polynomial function that does not have a quadratic or linear term would satisfy the relationship, but seems trivial." $\endgroup$ Commented Apr 18, 2018 at 19:37
  • $\begingroup$ @Deoxal The above works for any $\,g\,$, not necessarily a polynomial. For example $\,f(x)=e^{x^3}\,$ and $\,f(x)=\sin(x^3)\,$ both fall into this class of functions, while they are obviously not polynomials. $\endgroup$ Commented Apr 18, 2018 at 20:23
  • $\begingroup$ I understand what you are saying but I want no trivial specific example. I am having trouble generating one without a numerical method. $\endgroup$ Commented Apr 19, 2018 at 15:31
  • $\begingroup$ @Deoxal The second part of the answer gives the general form. Take for example $\,g(x)=\cos(x)-1\,$ then you get $\,f(x)=\sin(x)-x\,$ which satisfies the condition. You can generate as many non-trivial examples as you wish by choosing different $\,g$'s $\endgroup$ Commented Apr 19, 2018 at 15:58
  • $\begingroup$ I graphed the two functions you gave and yes they do satisfy the condition. However, I still consider it trivial. I realize I did not give a clear definition of what trivial means, so I will include that above as well as a link to a polynomial solution that I would consider non-trivial. $\endgroup$ Commented Apr 19, 2018 at 17:01

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