For a function $y = f(x)$, a number $x_0$ is called $\textit{critical}$ if either $f'(x_0) = 0$ or $f'(x)$ does not exist. Sometimes the term $\textit{stationary}$ is used, but it is by far less popular. My question is

Why is the word "critical" used in this case as terminology? What makes $x_0$ critical if $f'(x_0) = 0$?

Of course the tangent line being horizontal or not being able to draw a tangent line gives local minimums and maximums. So why are maximums and minimums "critical"? It seems that "stationary" is more appropriate. So I am puzzled as to why the latter is less popular.

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    $\begingroup$ Critical as in "important" or "key" (for analyzing the behavior of the function). $\endgroup$ – quasi Jul 6 '17 at 23:44
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    $\begingroup$ Why is a feline creature called a "cat"? $\endgroup$ – BruceET Jul 6 '17 at 23:44
  • $\begingroup$ 'Sometimes the term stationary is used, but it is by far less popular'. Hm.. I've seen 'stationary' used more myself. $\endgroup$ – Shuri2060 Jul 6 '17 at 23:45
  • $\begingroup$ @quasi Of course the are "important" and/or "key", but I don't think the word "critical" has the same meaning. $\endgroup$ – user 1987 Jul 6 '17 at 23:52
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    $\begingroup$ I think "critical" is used in the sense of a point where a situation changes. (as in "critical mass") Merriam Webster's first definition is "of realating to, or being a turning point or specially important juncture". As the is a point where a slope turns from positive to negative (or vice versa), it seems important to me. I don't know if I'm write but I accept it and it feels good to me. It's just language. And language is language. $\endgroup$ – fleablood Jul 6 '17 at 23:58

Notice the primary $1 a$ definition according to Merriam-Webster (the bolding are mine):

Definition of critical

1a : of, relating to, or being a turning point or specially important juncture a critical phase: such as (1) : relating to or being the stage of a disease at which an abrupt change for better or worse may be expected; also : being or relating to an illness or condition involving danger of death critical care a patient listed in critical condition (2) : relating to or being a state in which or a measurement or point at which some quality, property, or phenomenon suffers a definite change critical temperatureb : crucial, decisive a critical testc : indispensable, vital a critical waterfowl habitat a component critical to the operation of a machined : being in or approaching a state of crisis a critical shortage a critical situation

I think it is pretty apt, don't you?

  • $\begingroup$ Now, is this before or after the source mentioned by Robert Israel? $\endgroup$ – James S. Cook Jul 7 '17 at 0:11
  • $\begingroup$ @JamesS.Cook Just about to ask :) $\endgroup$ – user 1987 Jul 7 '17 at 0:12
  • $\begingroup$ Well, this is the latest version of the dictionary but obviously the english word came first. 1547 according to Webster's. $\endgroup$ – fleablood Jul 7 '17 at 0:13
  • $\begingroup$ We could go back to the Greek original - for example, Hippocrates uses κρίσις for the turning point of a disease in precisely the way described here $\endgroup$ – Hagen von Eitzen Jul 7 '17 at 11:06
  • $\begingroup$ Now I'm a little curious as to when criticism and critical as is judgemental came about. $\endgroup$ – fleablood Jul 7 '17 at 20:51

According to Earliest Known Uses of Some of the Words of Mathematics:

CRITICAL POINT is found in 1871 in A General Geometry and Calculus by Edward Olney [University of Michigan Historic Math Collection].

I'm somewhat surprised it is that recent, but I guess the systematic consideration of the possibility of non-differentiability didn't really happen before the mid-19'th century.

EDIT: As it turns out, Olney's book is available on-line and searchable. The first appearance of "critical" is on p. 118:

As $y = f(x)$, this problem is the ordinary one of maxima and minima of functions of a single variable, treated in the Calculus. Hence we find the values of $x$ which render $\dfrac{dy}{dx} = 0$ or $\infty$, as critical values, i.e., values to be examined, and at which the property exists, if it exist at all.

  • $\begingroup$ Thanks for the reference. I guess will never find out why Edward Olney made that choice or why agreed to with that choice :) $\endgroup$ – user 1987 Jul 6 '17 at 23:59
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    $\begingroup$ It seems very apt to me. Critical doesnt mean dangerous or essential nescessarily (although it has those secondary meanings) but rather means (primarily) a turning point where something may turn or go either way. In that meaning it is incrediably apt. $\endgroup$ – fleablood Jul 7 '17 at 0:02

Critical as in "important" or "key" (for analyzing the behavior of the function).

For a continuous function from $\mathbb{R}$ to $\mathbb{R}$, the critical points may or may not correspond to actual turning points, but they are the only places where a turning point is possible. Thus, one could say that analyzing the local behavior of the function at or near such points is critical to understanding the behavior of the function.

But note: The term "critical point" is not synonymous with the term "stationary point". All stationary points are critical, but not every critical point is stationary.

If $f\colon\mathbb{R}\to\mathbb{R}$ is continuous,

  • $x=c\;$ is a stationary point for $f$ if $f'(c)=0$.
  • $x=c\;$ is a critical point for $f$ if either $f'(c) = 0\;$ or $f'(c)$ does not exist.

I guess there might be some good explanations as to why we call them critical point, but the root of your question is kind of pointless (no offense) :

You gave the definition of a critical point, when we define a word, especially in mathematics, it's a way to allow us to better communicate our ideas, but that's it.

I could say : "Wow do you see those three guys down below ? They form a nice three sided shape." But I use : "(...) they form a nice triangle.". Because we, as humans, already agreed on the definition of a triangle. We could have called them potatoes or scissors, the word itself doesn't matter, as long as we all agree on the meaning.

Shall I say it is critical that you understand this ? =)

EDIT to adress the comments :

  • chepner is on point, most mathematics words have latin or greek roots. I guess it might be confusing for english or germanic speaking people as it is more often than not completely different to what they use. When fleablood says "secant" is obscure to him, I think it's a good example : I speak french and here we got words like "sécateur" (which is a kind of scissor for plants) so it's easy to understand secant.

  • The word "pointless" I used seems not to be ideal and you have to put this on my lack of vocabulary. Again I didn't mean to offend.

  • For me, I understand "critical points" as "important, defining points" for a function, like if I had to describe the graph of a function in a sentence (not with the equation), those were the points I would name

  • $\begingroup$ No offense taken. However, when we use words to describe mathematical objects, there should be a connection. Hopefully a clear connection. With your reasoning, even if such points were called $\textit{sexy}$, the question would not make sense. $\endgroup$ – user 1987 Jul 7 '17 at 0:08
  • $\begingroup$ And there usually are. Although it can sometimes be archaic and stuffy ("trancendental" is a bit high-faluting if you ask me.. but apt). Critical means at a stage were change is imminent. As a functions rate of change is likely to switch direction, it makes good sense to me. $\endgroup$ – fleablood Jul 7 '17 at 0:12
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    $\begingroup$ The question is not pointless. The word critical (along with many other mathematical terms) comes up in different contexts and its meaning in one setting often tells us what is meant in a different setting. $\endgroup$ – dannum Jul 7 '17 at 0:15
  • $\begingroup$ I don't think it is pointless but sometimes the origin con be obscure and even unknowable. And it won't affect the mathematics. "Secant" is a rather obscure one in my opinion. I also don't really understand "normal" (I sort of do but mostly don't). $\endgroup$ – fleablood Jul 7 '17 at 0:21
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    $\begingroup$ The Latin word for a carpenter's square was norma; something that was normalis was in accordance with a square/right angle. Broaden the concept to something in accordance with any external expectation, and we get our modern meaning of "normal". A secant line is one that cuts across a circle, intersecting with it twice, and "secant" is derived from Latin secare, "to cut". (A special case of a secant is a tangent, which touches--Latin tangere, "to touch"--a circle at exactly one point instead.) $\endgroup$ – chepner Jul 7 '17 at 15:53

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