value vs point

[Loki123]

What is the difference between critical value and critical point?
Is critical value x coordinate while critical point is (x,y)?

 

[Deleted member 4993]

What is the difference between critical value and critical point?
Is critical value x coordinate while critical point is (x,y)?

Can you provide the original problem where you encountered the term "critical value"?

 

[Loki123]

Can you provide the original problem where you encountered the term "critical value"?

there are no problems.
here are some definitions with it.
DEFINITION A critical value of a function f is any number c in the domain of f for which the tangent line at is horizontal or for which the derivative does not exist. That is, c is a critical value if exists and or does not exist.
that's page 201

https://www2.math.binghamton.edu/lib/exe/fetch.php/people/mckenzie/bittinger_et_al..pdf

 

[Deleted member 4993]

DEFINITION A critical value of a function f is any number c in the domain of f for which the tangent line at is horizontal or for which the derivative does not exist. That is, c is a critical value if f(c) exists and f'(c) = 0 or f'(c) does not exist

Although you did not quote the definition correctly, the definition is very clear. Where is your confusion?

 

[Dr.Peterson]

What is the difference between critical value and critical point?
Is critical value x coordinate while critical point is (x,y)?

I have seen both terms used, either interchangeably, or, as you suggest, with "point" used to refer to the pair and "value" to the x-coordinate.

Here is an example using "critical point" to refer to a single number (presumably thought of as a point on the number line):

Calculus I - Critical Points

In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions.

tutorial.math.lamar.edu

When your source gives a definition, use that definition. Words, even in mathematics, are not always defined in exactly the same way, even in the same country. It is not always useful to look for distinctions like this!

 

[Loki123]

I have seen both terms used, either interchangeably, or, as you suggest, with "point" used to refer to the pair and "value" to the x-coordinate.

Here is an example using "critical point" to refer to a single number (presumably thought of as a point on the number line):

Calculus I - Critical Points

In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions.

tutorial.math.lamar.edu

When your source gives a definition, use that definition. Words, even in mathematics, are not always defined in exactly the same way, even in the same country. It is not always useful to look for distinctions like this!

Okay got it. But one more question:
Would you say a critical point can be an extreme value
Or
A critical value can be an extreme value
Or
A critical point can be an extreme point

 

[Deleted member 4993]

Okay got it. But one more question:
Would you say a critical point can be an extreme value
Or
A critical value can be an extreme value
Or
A critical point can be an extreme point

Look at the definition of the critical point and tell us what do you think and why do you think so.

 

[Loki123]

Look at the definition of the critical point and tell us what do you think and why do you think so.

I would think that since point is (x, y) and value is just x that we'd have to either point or value, not mix them. However, I am not sure whether point or value is correct.

 

[AvgStudent]

I would think that since point is (x, y) and value is just x that we'd have to either point or value, not mix them. However, I am not sure whether point or value is correct.

Hi Loki,
Not all critical points/values are extreme points/values. A critical point can also be an inflection point which is not an extreme point/value.

 

[Dr.Peterson]

Okay got it. But one more question:
Would you say a critical point can be an extreme value
Or
A critical value can be an extreme value
Or
A critical point can be an extreme point

Here, you need to check the definitions of extreme value and extreme point. Have you? What definitions do you find?

I would say they are different. But as I said, you need to go by your source(s).

What others are saying is not relevant to your actual question; you are correctly not saying that a critical point is necessarily an extreme point. Your use of the word "can" is appropriate.

 

[Loki123]

Here, you need to check the definitions of extreme value and extreme point. Have you? What definitions do you find?

I would say they are different. But as I said, you need to go by your source(s).

What others are saying is not relevant to your actual question; you are correctly not saying that a critical point is necessarily an extreme point. Your use of the word "can" is appropriate.

From what I know
All extreme values are critical
But not all critical are extreme

 

[Dr.Peterson]

From what I know
All extreme values are critical
But not all critical are extreme

But that's not what I understand you to have asked. You asked about the distinction, if any, between "extreme point" and "extreme value"; and that requires seeing what definitions or statements you have found. As I've indicated, terminology varies (which means your question is not very important).

As I see it, an extreme value is a value of y: the highest or lowest value of the function. An extreme point might be either an ordered pair, or just the value of x, which tells where the extremum is.

The same source I referred you to before, in the next section uses the word "extrema" and mostly avoids either of the phrases you are asking about. But it does say, "As this example has shown there can only be a single absolute maximum or absolute minimum value, but they can occur at more than one place in the domain," which clearly refers to y. For an example where "extreme point" refers to x, see here (page 6): "Points a where f has a maximum or a minimum are called extreme points of f ."

 

[Loki123]

But that's not what I understand you to have asked. You asked about the distinction, if any, between "extreme point" and "extreme value"; and that requires seeing what definitions or statements you have found. As I've indicated, terminology varies (which means your question is not very important).

As I see it, an extreme value is a value of y: the highest or lowest value of the function. An extreme point might be either an ordered pair, or just the value of x, which tells where the extremum is.

The same source I referred you to before, in the next section uses the word "extrema" and mostly avoids either of the phrases you are asking about. But it does say, "As this example has shown there can only be a single absolute maximum or absolute minimum value, but they can occur at more than one place in the domain," which clearly refers to y. For an example where "extreme point" refers to x, see here (page 6): "Points a where f has a maximum or a minimum are called extreme points of f ."

Oh okay. Got it. Thank you so much