Critical Pts (2 Viewers)

[Lukybear]

How are they defined?

I had ideas that they were undefined and hence were critical pts. But also, i saw in Patel, that relative maxima, relative minima, and inflexion pts are criticals. Expand anyone?

 

[GUSSSSSSSSSSSSS]

critical points are any time that the graph changes (crap definition but yeaaaa lol)

so this includes stationary points, and also as u said: discontinuities, cusps, etc

 

[Cazic]

A critical point of a function f is any point x in the domain of f for which the first derivative f' at the point x is 0 or not defined? I wouldn't call an inflection point a critical point (unless it's a horizontal inflection point), so don't take my word for it, take your text-book's/teacher's word for it.

 

[The Nomad]

Really...a definition is useless.

 

[Cazic]

Yes, after a comment like a definition is useless, only to follow by a (not so helpful) definition, you're missing an invitation to leave the mathematics forum for the mathematical crimes of (1) disparaging definitions, and (2) abuse of logic

I'm happy to oblige.



Joking .. mostly

 

[Trebla]

The syllabus definition is when the first derivative is undefined or zero.

An inflexion point would be a critical point of the first derivative function as opposed to the original function.

 

[Cazic]

Surely they say something about the domain of the original function? They don't consider 0 a critical point of 1/x do they?

 

[cutemouse]

Surely they say something about the domain of the original function? They don't consider 0 a critical point of 1/x do they?

x=0 or y=0

Of course not because it's an asymptote.

 

[MongMan]

The syllabus definition is when the first derivative is undefined or zero.

This. Take first derivative. then solve x for dy/dx = 0

 

[The Nomad]

Yes, after a comment like a definition is useless, only to follow by a (not so helpful) definition, you're missing an invitation to leave the mathematics forum for the mathematical crimes of (1) disparaging definitions, and (2) abuse of logic

I'm happy to oblige.



Joking .. mostly

Previous post edited for your sake.

 

[annabackwards]

y' = 0 or y' is undefined

POIs are only critical points if they're horizontal POIs

 

[Cazic]

Cmon. Is -2 a critical point of sqrt(x)?

Edit: This thread reminds me of the claim that 1/x is discontinuous at x=0

 

[cutemouse]

Cmon. Is -2 a critical point of sqrt(x)?

Well no, because the second derivative would not be undefined or zero, as per the definition Trebla provided.

It simply would not exist.

 

[cyl123]

Um i think the definition of a function should be raised.

From Wolfram:
A function is a relation that uniquely associates members of one set with members of another set.
A function y=f(x) has critical points at all points x0 where f^'(x0)=0 or f(x) is not differentiable.

So f: R -> R (real numbers in domain mapping to real numbers in codomain), where f(x)=1/x is not a function as f(0) is not defined. Thus f does not uniquely associate 0 with a real number in the codomain and f is not a function, thus we cannot use the definition of critical points for a function on f. Thus we cannot say 0 is a critical point.

However, if f: R/{0} -> R (real numbers excluding 0 in domain mapping to real numbers in codomain) where f(x)=1/x is a function, as every real number in the domain can be mapped to a real number in the codomain. Thus the definition of critical points for a function can be applied. However, 0 is not in the domain of f, so it cannot possibly be a critical point.

Also, for f: R--> R, f(x)=sqrt(x) is not a function, as it does not map negative numbers to any numbers in the real codomain, however for g: (0, inf) --> R, g(x)=sqrt(x) and h:Real --> Complex where h(x)=sqrt(x) are functions.

EDIT: Yeh but for high school definitions are useless, and sometimes wrong...

 

[Cazic]

Um i think the definition of a function should be raised.

From Wolfram:
A function is a relation that uniquely associates members of one set with members of another set.
A function y=f(x) has critical points at all points x0 where f^'(x0)=0 or f(x) is not differentiable.

Implicit here is that x is in the domain of f.

So f: R -> R (real numbers in domain mapping to real numbers in codomain), where f(x)=1/x is not a function as f(0) is not defined. Thus f does not uniquely associate 0 with a real number in the codomain and f is not a function, thus we cannot use the definition of critical points for a function on f. Thus we cannot say 0 is a critical point.

However, if f: R/{0} -> R (real numbers excluding 0 in domain mapping to real numbers in codomain) where f(x)=1/x is a function, as every real number in the domain can be mapped to a real number in the codomain. Thus the definition of critical points for a function can be applied. However, 0 is not in the domain of f, so it cannot possibly be a critical point.

Nice.

Also, for f: R--> R, f(x)=sqrt(x) is not a function, as it does not map negative numbers to any numbers in the real codomain, however for g: (0, inf) --> R, g(x)=sqrt(x) and h:Real --> Complex where h(x)=sqrt(x) are functions.

Your function h is not well-defined without choosing a branch of sqrt map. Otherwise, nice (assuming you apply the same observation about the domain/critical points as you did above).

EDIT: Yeh but for high school definitions are useless, and sometimes wrong...

They definitely aren't useless, and definitions can't be wrong - they're definitions. They should be consistent with what is likely to be seen in later work though.

 

[cyl123]

I agree that definitions are important, except high school maths is currently not rigorous enough to deal with them in detail (for example defining functions without specifying domain and codomain). High school mathematics is just applications, where proving important theorems is rare. Proofs are where definitions need to be well known but unfortunately proofs are not a big part of the syllabus. (I dont mean proving results via LHS and RHS, I mean actual proofs of theorems from definitions and axioms).

Btw, I assumed that i took the positive branch of the square root map.

 

[Cazic]

Domain and range is covered at high school, and I think you'll find definitions are more prevalent than you let on. Textbooks aren't written in a Definition / Theorem / Proof style, sure, but I'm sure most of the detail is there.

Btw, there is no concept of "positive" when dealing with complex numbers. Perhaps you meant the upper half plane without the negative real line?

 

[cyl123]

Oops....sorry you're right about the complex numbers. probably should redefine the codomain as the portion of complex numbers z where Re(z)>=0.

Yes, the definitions are provided to students via textbooks, but knowing them rigorously is not needed for most part of their examinations and hence the rigor of definitions are often ignored.

 

[Cazic]

I would say



or



or whatever, with the obvious definition (too lazy to type it out)