A rigorous yet intuitive summary of inflection and critical points for beginning calculus?

Question

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 $f'(x)=0$.

   • 1.1. Example to illustrate the 'interior' part of Definition (1): The function $f: A \to \mathbb R, f(t)=t^2$ does not have a critical point at $x=0$ if $A \in \{(-\infty,0], [-7,-3) \cup [-2,0], [0,4], [-8,-6) \cup \{0\} \cup (2,10),\{0\},\{0,1\} \}$ (rigorously, all these $A$'s are given the subspace topology...or the topology of $\mathbb R$...not sure...which is the right one?).

   • 1.2. An example for the 'undefined' part of Definition (1) $f:\mathbb R \to \mathbb R, f(t)=|t|$

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.

   • 5.1. A counterexample to the converse of Consequence (4) for $f''(x)=0$ is $f: \mathbb R \to \mathbb R, f(t)=\frac{t^4}{12},f''(t)=t^2$, where $x=0$ is an undulation point rather than an inflection point.

   • 5.2. A counterexample to the converse of Consequence (4): When is $f''(x)$ undefined but $x$ not an inflection point of $f$?

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 $(x,f(x))$.

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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.