# Functions where critical numbers equal points of inflection

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.

 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.

• Thanks that was what I was looking for. – Deoxal Apr 19 '18 at 21:44

## 1 Answer

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 $\;n_1, n_2, \ldots \ge 3\;$

• 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." – Deoxal Apr 18 '18 at 19:37
• @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. – dxiv Apr 18 '18 at 20:23
• I understand what you are saying but I want no trivial specific example. I am having trouble generating one without a numerical method. – Deoxal Apr 19 '18 at 15:31
• @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 – dxiv Apr 19 '18 at 15:58
• 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. – Deoxal Apr 19 '18 at 17:01