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Let $f(x): [a,b] \to \mathbb{R}$ be a twice differentiable non-constant function on $[a,b]$. Let $\alpha \in (a,b)$ be the abscissa of an inflection point of $f(x)$. Is it necessary that the $f(x)$ is either non-increasing or non-decreasing at $x = \alpha$?

My gut feeling is that the statement is true, because thinking about inflection points and change in curvature it always seems intuitive that the function is increasing or decreasing. For some context, I was reading an example in my textbook which asked to find the points of extrema for a polynomial function. One critical point was a stationary point, and this was also a point of inflection. The example concluded "$x$ is not a point of extremum since the concavity is changing in the neighbourhood of a stationary point", without any further explanation which also made me think about non-stationary points of inflection.

I don't know where to start on proving such a thing, although in my limited experience I have not come across any counter-example.

Edit: A function $f$ is non-decreasing or non-increasing at a point $x$ if $f(t) \geq f(x)$ for every $t > x$ close enough to $x$ and $f(t) \leq f(x)$ for every $t < x$ close enough to $x$, or $f(t) \leq f(x)$ for every $t > x$ close enough to $x$ and $f(t) \geq f(x)$ for every $t < x$ close enough to $x$. This answer covers a similar definition more rigorously.

Box Box Box Box
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    How do you define monotonicity at a point? – Ken Hung Jul 06 '21 at 08:19
  • @KenHung By monotonicity at a point, I'm referring to monotonicity in a small interval around that point. Another way of saying the same thing would be that the function is monotonic at a point if it is non-increasing or non-decreasing at the point. – Box Box Box Box Jul 06 '21 at 08:47
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    I think it would be better if you provide this definition in the post to avoid confusion. – Ken Hung Jul 06 '21 at 08:48
  • @KenHung ok, I have done that. I thought this was a standard definition, although it seems like it isn't. – Box Box Box Box Jul 06 '21 at 08:51
  • Also a follow up on the definition you have provided, by small interval do you mean arbitrary small interval, meaning that one can choose the length of the interval? – Ken Hung Jul 06 '21 at 08:52
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    I think in your proof you have to be careful about the fact that if $\alpha$ is an inflection point of $f \in \mathcal C^2$, then $f''(x)$ in $\alpha$ has an isolated zero and changes sign.

    If you have a non isolated zero of the second derivative the function might not be locally monotone in $\alpha$.

    Consider for example the function $$f(x) = \begin{cases} x^5\left(1-\sin\frac1x\right) & (x\neq 0)\ 0 & (x=0). \end{cases} $$

     – dfnu Jul 06 '21 at 10:43

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