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I haven't done these in awhile. While my analysis covered continuity but not differentiability, I have so far not revisited these in learning geometry or algebra. I am trying to help a calculus student, so any notion of 'interior' is intuitive.

First, please verify if these are correct. I think I may be missing phrases like 'open neighbourhood' or 'open interval' and might have confused my ifs and only ifs. Some collections like Wikipedia, are mixed for both higher maths and beginning calculus.

Let $A, B \subseteq \mathbb R$.

  1. Definition of critical point: A function $f: A \to B$ has a critical point at $x \in A$ (I represent the point $(x,f(x))$ by the real number $x$) if (a) $x$ is in the interior of $A$ and (b) $f'(x)$ is undefined or $f'(x)=0$.

  2. Definition of inflection point: A function $f: A \to B$ has an inflection point at $(x,f(x))$, where $x \in A$ may be at the boundary of $A$ or the interior of $A$ but must be in $A$ (I believe this is both intuitive and rigorous: A point in a set is either in the set's interior or the set's boundary), if $f$ has a tangent line at $(x,f(x))$ and the concavity of $f$ changes at $(x,f(x))$.

    • 2.1. Equivalent definition of inflection point: (This is my attempt to try to rigorously explain 'has a tangent line' while incorporating tangent lines with infinite slope like in this example with $f(t)=t^{1/3}$) A function $f: A \to B$ has an inflection point at $(x,f(x))$ if $f'$ exists in $C$, the intersection of the whole of $A$ and not just $A$'s interior with some open interval $(a,b)$ ($x \in C = A \cap (a,b)$) where (a) $f'$ is increasing on $(\inf C,x)$ and not increasing (constant, decreasing or does not exist) on $(x,\sup C)$ or (b) $f'$ is decreasing on $(\inf C,x)$ and not decreasing (constant, increasing or does not exist) on $(x,\sup C)$.
  3. Equivalent definition of an inflection point when $x$ is an interior point (for the previous example of $f(t)=t^{1/3}$, I assume the concern is that $x=0$ is not an interior point): A function $f: A \to B$ has an inflection point at $x$, where $x$ is an interior point of $A$, if $f'$ exists in some open interval $(a,b)$ that both contains $x$ and is contained in $A$ ($x \in (a,b) \subseteq A$, and $(a,b)$ is contained in $A$ by the definition of $x$ as an interior point of $A$), where (a) $f'$ is increasing on $(a,x)$ and decreasing on $(x,b)$ or (b) $f'$ is decreasing on $(a,x)$ and increasing on $(x,b)$.

  4. Consequence of the definition of inflection: A function $f: A \to B$ has an inflection point at $x \in A$ only if $f''(x)$ is undefined or $f''(x)=0$.

    • 4.1. An example of Consequence (4) for the 'undefined' part is based precisely on Example (1.2) $f: \mathbb R \to \mathbb R, f(t) = \begin{cases} -\frac{t^2}{2} &\text{if $t < 0$} \\ \frac{t^2}{2} &\text{if $t \geq 0$.} \end{cases}, f'(t)=|t| $.
  5. Observation: The 'undefined or zero part' of Consequence (4) is just like in Definition (1) except (4) is not a definition for inflection.

  6. Proposition: A differentiable function $f: A \to B$ has an inflection point at an interior point $(x,f(x))$ only if $f'$ has a critical point at $(x,f(x))$.


Second, I use the above to rigorously (as rigorously as possible for calculus students) answer as follows. Please verify. If an answer or argument (such as if something in the first part above is wrong) is incorrect, then please give the corresponding correct answer or argument.:

Find critical points and inflection points given the derivative (also check for consistency with Darboux's theorem)?

I am splitting this up to not cover a lot in one post.

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    $\begingroup$ One thing you have to keep in mind is that authors vary in what "critical point" and "inflection point" mean. Regarding inflection point variations by authors, see this 28 December 2005 ap-calculus post archived at Math Forum and the 4 papers I cited in this comment. Incidentally, for ordinary beginning calculus in the U.S. sense, you're going way into left field with the student. $\endgroup$ – Dave L. Renfro Feb 23 at 8:51
  • $\begingroup$ @DaveL.Renfro Thanks! Did I get the interior parts correct please? In particular Example (1.1) and Definition (3). $\endgroup$ – user636310 Feb 24 at 2:57
  • $\begingroup$ I would restrict $A$ (domain set) of the definitions to nondegenerate intervals (intervals that are not singleton sets or the empty set), and as for what definition to use for "inflection point", that's something I've said varies with author, so what you need to do is to carefully look at the textbook of the person you're tutoring and stick to the version used there. $\endgroup$ – Dave L. Renfro Feb 24 at 7:18
  • $\begingroup$ @DaveL.Renfro I mean, Definitions (1) and (2) are given by the author, so now I would like to know if everything else follows from those definitions. From a pedagogical point of view, such an $A$ would be ideal for basic calculus or at least nondegenerate intervals or finite disjoint unions of them, but part of this question is to see for counterexamples that would be unsuitable for basic calculus and therefore the instructors should change their assumptions for $A$. $\endgroup$ – user636310 Feb 24 at 16:33
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    $\begingroup$ I would advise not getting mired in many of these technicalities, since definitions used in elementary calculus courses are sometimes only as precise as needed and no more, and for the student you're trying to help, these issues are too much of a diversion. Thus, for critical points, the non-differentiable points of interest basically arise in two ways for the kinds of explicit-formula functions dealt with: corner points, such as arise when absolute values are involved, and where the derivative becomes infinite, such as arise when a variable's exponent is strictly between $0$ and $1.$ $\endgroup$ – Dave L. Renfro Feb 24 at 18:30

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