Critical points

[HallsofIvy]

This is correct, but the statement ought to read, "x=0 and x=2 as the critical points".

Critical points are values of x, not f(x).

Strictly speaking, critical points are points and so are neither values of x or f(x) but \(\displaystyle (x, f(x))\)

 

[Quaid]

Strictly speaking, critical points are points and so are neither values of x or f(x) but \(\displaystyle (x, f(x))\)

Are you discussing something beyond introductory calculus? This perspective is not how the subject is taught, at the introductory level. (One may confirm, by googling for the definition.)

If you would like to claim that x=2 is not a critical point, but that (2,f(2)) is, then what is your value for the y-coordinate? :?

 

[lookagain]

Are you discussing something beyond introductory calculus?
No, HallsofIvy is not discussing something beyond introductory calculus.

This perspective is not how the subject is taught, at the introductory level.
You are incorrect.

(One may confirm, by googling for the definition.)
Any definitions that oppose the examples at the links below are incorrect.

If you would like to claim that x=2 is not a critical point,
It's not just a claim, it's a fact. It is called a "critical number."

but that (2,f(2)) is, then what is your value for the y-coordinate? :?

The x-values are critical numbers.

Critical numbers are discussed here, for example:

http://www.analyzemath.com/calculus/applications/critical_numbers.html



The xy-coordinates (points) are called "critical points."

Critical points are discussed here, for example:

http://www.cliffsnotes.com/math/calculus/calculus/applications-of-the-derivative/critical-points

 

[HallsofIvy]

Are you discussing something beyond introductory calculus? This perspective is not how the subject is taught, at the introductory level. (One may confirm, by googling for the definition.)

If you would like to claim that x=2 is not a critical point, but that (2,f(2)) is, then what is your value for the y-coordinate? :?

The "point" of my response was that a critical point is a [/b]point[/b], not a number! What is your justification for saying that the number "2" is a point?

 

[Quaid]

The x-values are [called] critical numbers.

Okay, lookagain -- I'm not familiar with that terminology. Is this distinction important, at the introductory level? I just don't see it used.

Let's google keywords definition critical point calculus

At the top of the results, I see a definition insert. It says, "…critical point of f is x = 0."

Let's check the first link (Paul's Online Notes). It says, "…x=c is a critical point of the function f(x)…"

Let's check the second link (Wikipedia). It says, "…f(x) has a critical point x₀ with critical value y₀…"

This is the language that I've seen in introductory courses, texts, lectures, and material.

Maybe, it's another case of mushy communication in mathematics. But, as it seems to me, a majority refer to values of x when talking about critical points, so I do also.

Regarding the OP, the number 2 is not even in the domain of function f; the point (2,f(2)) does not exist. Even so, I'm confident that most people would say, "x=2 is a critical point" because f'(2) does not exist.

 

[Quaid]

What is your justification for saying that the number "2" is a point?

You didn't respond to my questions, but I'll answer yours.

My justifications are the dimensionless points which comprise the Real number line.

 

[lookagain]

Okay, lookagain -- I'm not familiar with that terminology. Is this distinction important,
at the introductory level?
Regardless of the level it is important. If the problem is to find a "critical point" for the graph,
then they should have the expectations of reporting (x, y) answers.

I just don't see it used.Let's google keywords definition critical point calculus
At the top of the results, I see a definition insert.

It says, "…critical point of f is x = 0."Let's check the first link (Paul's Online Notes).
It says, "…x=c is a critical point of the function f(x)…"Let's check the second link (Wikipedia).
It says, "…f(x) has a critical point x₀ with critical value y₀…"

I don't use popular results as the ones above to gauge what I teach/tutor the students in.

This is the language that I've seen in introductory courses, texts, lectures, and material
.Maybe, it's another case of mushy communication in mathematics.
But, as it seems to me, a majority refer to values of x when talking about critical points, so I do also.

The definitions given for googling are mixed, and with the large sample I looked at,
the majority reflect the x-value. However, your reasoning to "follow the herd"
because of that is not a logical justification.

Regarding the OP, the > > > number 2 < < < is not even in the domain of function f;
the point (2,f(2)) does not exist. Even so, I'm confident that most people would say,
"x=2 is a critical point" because f'(2) does not exist.

1) I would expect students to parrot their instructors and state it this way.
2) I noticed you used the phrase "number 2." I would play off that then and
state that "x = 2 is a *critical number*."

Here are a minority of the samples I looked across that I subscribe to:


www.chegg.com/homework-help/definitions/critical-point-29

http://www.chegg.com/homework-help/definitions/critical-point-29

http://www.mathwords.com/c/critical_point.htm

http://www.merriam-webster.com/dictionary/critical point

http://education-portal.com/academy...s-in-calculus-function-graph-quiz.html#lesson

http://pblpathways.com/calc/C12_1_2.pdf

https://bccalculus.wikispaces.com/D...maximum+and+minimum+values,+with+optimization

http://mathforum.org/mathimages/index.php/Critical_Points

http://www.algebra.com/algebra/home...ogarithmic-functions.faq.question.414181.html

 

[Quaid]

Hi lookagain. I agree with you, in that exercises which ask for a specific point on a graph require an ordered pair to describe it.

How would you state the critical points, in the OP? Is there only one, from your point of view?

 

[lookagain]

How would you state the critical points, in the OP? Is there only one, from your point of view?

Yes, it's the point (0, 0), which happens to also be the absolute minimum point. (I graphed it on the Internet.)