Why is this not a local maximum?

[PhizKid]

Homework Statement

Why is 'a' not a local maximum?

Homework Equations

The Attempt at a Solution

According to the definition of a local maximum, I have to take an open interval around 'a' so it seems like 'a' is a local maximum. I don't understand why it isn't.

 

[Ray Vickson]

Homework Statement

Why is 'a' not a local maximum?

Homework Equations

The Attempt at a Solution

According to the definition of a local maximum, I have to take an open interval around 'a' so it seems like 'a' is a local maximum. I don't understand why it isn't.

Definitions vary somewhat. If the domain you are interested is only a finite interval I, an end-point such as x = a is often regarded as a local maximum, since it satisfies the definition that f(a) >= f(x) for all x in I, |x-a| small. Typically (although not always), optimization textbooks would use this type of definition, since it fits exactly with practical needs (which is to find the best value of the function, no matter where it happens to be located). However, general math textbooks might use a different definition. So, the best advice is: go with the definitions used by your instructor and/or textbook.

RGV

 

[PhizKid]

I am just severely confused because the textbook states that the right endpoint on this graph is a maxima as well, but the left endpoint is not.

 

[I like Serena]

The right end point is an absolute maximum.
The left end point is a local (but not absolute) maximum.

 

[PhizKid]

Okay, so the textbook has an error then, I guess.

 

[I like Serena]

Okay, so the textbook has an error then, I guess.

Yes, unless the domain is not limited on the left side at a.

 

[Ray Vickson]

Okay, so the textbook has an error then, I guess.

If the book meant "global maximum" then what it says is correct. Sometimes, a book will drop the adjective "global", so maybe that is what it did. (In that case, though, it might have to add the adjective "local" in some cases, to distinguish between the different types.)

RGV

 

[PhizKid]

The textbook asked to identify all the maximums and minimums, including local and absolute.

 

[SammyS]

The textbook asked to identify all the maximums and minimums, including local and absolute.

Assuming the right hand end point is b, is the function defined on [ a, b ], or is it defined on ( a, b ] ?

 

[I like Serena]

Assuming the right hand end point is b, is the function defined on [ a, b ], or is it defined on ( a, b ] ?

Interesting distinction! ;)

According to the definition on wiki there is no local maximum at a on (a,b].
However, every practical application shouts that there is and there should be.
Since there is no specific reference to this situation in wiki, in my book that means that wiki's definition is flawed.

Google does not help much, and for instance wolfram mathworld does not even give a proper definition.

 

[Ray Vickson]

Interesting distinction! ;)

According to the definition on wiki there is no local maximum at a on (a,b].
However, every practical application shouts that there is and there should be.
Since there is no specific reference to this situation in wiki, in my book that means that wiki's definition is flawed.

No: Wiki is correct. Take an easier example: what is the minimum of f(x) = x in the region x > 0? Answer: there *isn't* one. There is an "infimum", but not a minimum. If you don't believe this, please tell me what the minimum is.

RGV

 

[I like Serena]

No: Wiki is correct. Take an easier example: what is the minimum of f(x) = x in the region x > 0? Answer: there *isn't* one. There is an "infimum", but not a minimum. If you don't believe this, please tell me what the minimum is.

RGV

I did not say that wiki was wrong (yet).
But it should have addressed this issue and it didn't and in that respect it is flawed.
Perhaps we can fix it. :)

The distinction between an infimum and a minimum is a fine one and invites lots of discussion on what a word means exactly and when it should be used.

As for your question, I could say that the minimum is the infimum in that case, but you'll probably disagree and I'm not able to support it with references (yet).
Can you?

 

[Ray Vickson]

I did not say that wiki was wrong (yet).
But it should have addressed this issue and it didn't and in that respect it is flawed.
Perhaps we can fix it. :)

The distinction between an infimum and a minimum is a fine one and invites lots of discussion on what a word means exactly and when it should be used.

As for your question, I could say that the minimum is the infimum in that case, but you'll probably disagree and I'm not able to support it with references (yet).
Can you?

I could, but I won't bother, as the definitions are rather standard: a (local or global) minimum is a (local or global) infimum that is *attained*---that is, there is some allowed value of the variable that gives the infimum. For an open interval, and endpoint is not an "allowed" value. In practical terms the distinction is not that important, because if the infinum is 1 (say) and is not attained, in a practical application we could use instead an attainable value that gives, say 1.00000000000000000000000000000000000000001. But, in theory, there is always going to be a difference.

RGV