Can inflection points be determined from the local extrema of the first derivative? The local extrema of the first derivative determining the inflection points makes sense; for $x = c$ to be a local extremum for the first derivative, the first derivative's derivative (i.e., the second derivative) would need to change signs as it passes through $x = c$ and $f''(c)=0$. Wikipedia seems to agree:

the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x

The reason I ask is because of a multiple choices question from my highschool math book:

Which statement verifies that the function $f$ has a point of inflection at $x = c$?

The two answers I was stuck on were

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*The derivative $f'$ has a local maximum at $x = c$

*The sign of $f''$ changes at $x=c$
The answer is 2, but I don't really understand why 1 isn't correct. Maybe because inflection points can also happen if the first derivative has local minima? Though with the way the question is worded this shouldn't matter.
 A: I would think that statement 1. is incorrect for the reason you suggested: since inflection points also occur at local minimums of the first derivative, statement 1. does not wholly verify whether a point is an inflection point.
I will agree that this is not necessarily the best way to word the question, but statement 2. encompasses more cases (both local minimum and maximum), which makes it the better answer—and that's just how it goes with multiple-choice questions.
A: The original question is not a simple issue even though it originates from a high school textbook.

*

*If we adopt the "loose" sense of local minimum, ($f(x)\ge f(c)$ for all $x$ in a neighborhood of $c$, For example Stewart's Calculus) then one can easily come up with an example of $f$ for which the choice #1 does not verify the presence of inflection point at $c$. If $f$ is any linear function, then $f'$ is a constant. A constant function has a local minimum everywhere.


*Now, to make the issue more interesting, let's use the "strict" sense of local minimum ( $f(x)>f(c)$ for all $x\ne c$ in a neighborhood) In this case, it is much harder to come up with an example of a function $f$ such that $f'$ has a local minimum at $c$ yet $f$ has no inflection at $c$. One example is
$$f(x)= \int_0^x (2|t| + t \sin (1/t))\  dt$$This function and its derivative are shown here.


*Independently of the original posting, I would like to raise a question: Suppose $f$ is differentiable at $c$. Is the condition #1 (i.e. $f'$ having local extremum), a necessary condition for $f$ to have an inflection at $c$? This question appears to be harder than the "sufficiency" question raised in the original post and this question is also associated with one of the high school calculus exercise problem!
