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I've been out of a math program for about four years now. My wife is starting a CS degree, and finished her first calculus course last semester.

I tutored calculus throughout my entire undergrad, and have read quite a few calculus and real analysis textbooks. When I'm referring to a candidate for an $x$-value at which a point of extremum is located, I've always called it a critical point.

But my wife's professor taught three different terms:

  • Critical number: the $x$-value at which the possible extremum is located
  • Critical value: the corresponding $y$-value of the critical number
  • Critical point: a pair $(\text{critical number}, \text{critical value})$.

I don't understand the pedagogical value in creating these three terms.

  1. This doesn't look like standard terminology. I was confused at first when I was trying to help my wife learn about critical points.
  2. "Critical values" are rarely used in what I specialize in: statistics. I care more about where the extrema are possibly located, rather than what the actual $y$-values are at these possible candidates of extrema. And I like dropping constants whenever possible when it comes to having to find points of extrema, so the $y$-value is useless.

Can someone provide some rationale for why one would want to distinguish between these three bolded terms above?

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    $\begingroup$ I'm always surprised that people are aggrieved at having clear names for things. The language looks fairly standard; e.g., "critical number" and "critical point" are defined the same way in Stein/Barcellos Calculus and Analytic Geometry (1992). $\endgroup$ – Daniel R. Collins Jul 24 '18 at 18:05
  • $\begingroup$ (although OpenStax Calculus indeed uses "critical point" for the x-value.) $\endgroup$ – Daniel R. Collins Jul 24 '18 at 18:06
  • $\begingroup$ To many engineers, "critical point", "critical pressure", and "critical temperature" are important concepts. I am not sure they match what is described in this question, although they fit in the same context. $\endgroup$ – Jasper Jul 25 '18 at 0:34
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    $\begingroup$ I disagree that critical values are little used in statistics: the mean can be defined as the minimum ($x$) value of $E(d(x,X)^2)$ and the variance as the value at that minimum. $\endgroup$ – Adam Jul 26 '18 at 12:41
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    $\begingroup$ This seems traditional in higher mathematics: "If $df$ is singular, then $x$ is called a critical point of $f$, and the image $f(x)$ is called a critical value"; see Bott/Tu, Diff. Forms in Alg. Top. (1982); Milnor, Top. from a Diff. Viewpoint (1965); Sard's Theorem (1942); Morse, Calculus of Variations in the Large (1934); etc. But Granville's 1904 calc book defines critical value to be the value of $x$ in the domain of $f$, but I doubt the calc course was invented for mathematicians. $\endgroup$ – user1527 Jul 27 '18 at 3:34
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One rationale for introducing terms for these three objects is to encourage students to be clear about whether they are describing the input, output or pair $(x, y)$ for a function, and giving them language to do so efficiently.

Consider the question "Where is the maximum of the function $f$ on the given domain?" I've seen this type of wording trip students up, particularly if the back of the book just says "$5$". It's totally unclear (to me) whether the directive is to find the maximum output value, the input corresponding to that maximum output, or possibly the ordered pair. Giving names to these three things avoids a wordier rephrasing such as "At what x-value is $f(x)$ maximized on the given domain?"

Now, while my immediate reaction to introducing non-standard terminology is not positive, I do respect an attempt to get students to exercise care in how they communicate their ideas -- and one good way is by giving them language to do so. However, I think even more care should be exercised when introducing any subject-specific terminology, and new or non-standard terminology should almost never be given formally to a class.

[Note: On my shelf, Hughes-Hallett, Taalman, Anton and Faires all use "critical point" for the $x$-value, while two of them mention "critical value" and "stationary point". Sherwood and Taylor (c 1942) use "critical point" for the $(x,y)$ pair and "critical value" for the $x$-coordinate.]

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The standard terminology in use among mathematicians is to call a critical point a point $p$ where the differential of a differentiable function $f:M \to \mathbb{R}$ vanishes ($M$ is a smooth manifold, for example an open subset of $\mathbb{R}$), and to call a critical value the value $r$ it assumes at such a point.

The proposed terminology, calling $p$ a critical number only works in the limited context where $M$ is an open subset of $\mathbb{R}$ or $\mathbb{C}$, so can properly be called a number. It is confusing to call the pair $(p, r)$ a critical point for another reason. The point $p$ and the value $r$ "live" in different spaces. The pair $(p, r)$ lives in $M \times \mathbb{R}$, while $p$ lives in $M$ and $r$ lives in $\mathbb{R}$. In the special case where $M$ is an open subset of $\mathbb{R}$, then $(p, r)$ can be viewed as a point in $\mathbb{R}^{2}$, but this seems to me to generate unnecessary confusion, because the roles of the two coordinates are not symmetric.

On the other hand, one certainly wants to distinguish between the point at which a function is critical and the value it assumes there. This is just formalizing the distinction between the day of the year on which the coldest temperature occurs and what that temperature is, or the distinction between the point on the earth where the elevation above sea level is highest, and what that elevation is. One needs different names for different things.

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  • $\begingroup$ One doesn't even need to go to the generality of manifolds to see a problem with the term "critical number"; if the domain is 2- or 3-dimensional Euclidean space (as in Calc 3) the gradient doesn't vanish at numbers but at points. $\endgroup$ – Adam Jul 26 '18 at 12:49
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In primary, secondary, and some basic undergraduate courses, a point is an $n$-tuple (an ordered pair or an ordered triple), that is, a point is not a number. For a function of a single variable $y=f(x)$, we refer to $(x,y)$ as the point with abscissa $x$ and ordinate $y$.

So I can imagine that it would be confusing for these students if, given $(x_0,f(x_0))$, one refers to $x_0$ as a point. It would make more sense for them to call $(x_0,f(x_0))$ a point.

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    $\begingroup$ Precisely the point. $\endgroup$ – James S. Cook Jul 27 '18 at 1:33
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I think all three terms have value because they allow us to make important distinctions which would be muddled if we simply used the same term for all three. I'll focus this answer for a function $f$ on $\mathbb R$; that is, a real-valued function of a real-variable.

  1. critical number a number $x_o$ is critical if either $f'(x_o)$ does not exist or $f'(x_o)=0$. Notice, I do not assume $x_o$ is even in the domain of $f$. This becomes important if we analyze functions which have vertical asymptotes. It is possible for a function to change from increasing to decreasing at a point outside the domain of $f$. To give a complete account of the increase and decrease, concavity etc. it is necessary to consider critical numbers both inside and outside the domain (of course, for concavity we also must consider numbers for which the second derivative exists and is zero).
  2. critical value a number $y_o$ is called a critical value if it is the output of $f$ at a critical number in the domain of $f$; $y_o = f(x_o)$.
  3. critical point if $x_o$ is a critical number in the domain of $f$ then $(x_o,f(x_o))$ is a critical point.

All of this said, I don't usually penalize students for confusing the terminology. Honestly, I'm just happy if they get any of the above via correct calculus and analysis.

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