# Thread: Why is this not a local maximum?

1. ## Why is this not a local maximum?

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.

2. ## Re: Why is this not a local maximum?

It is a local maximum, just as you suspect.

The definition you're using that claims that it isn't a local maximum might have been misworded, or worded correctly but subtly in a way that you've overlooked. The important thing is that you've obviously understood what's actually meant by a local maximum.

(That you've detected this discrepancy means that you're engaged - thinking, questioning, and reflecting rather than just passively absorbing the material. I'd wager that you're a very good student.)

At the boundary of a domain, you consider the open interval intersect the domain. (Technically, that's what you're doing everywhere, except that distinction between open interval and open interval intersect the domain doesn't have any tangible consequences at most points in the domain.)

Local maximum (minimum) means that, when resticted to domain values that are some fixed distance from the domain value of the maximum (minimum), that the function's value is greater than or equal to (less than or equal to) all other values of the function in that region of the domain. It means that "nearby, the function is never greater (lesser)."

3. ## Re: Why is this not a local maximum?

I was very confused because the textbook states that it is not a maxima, but yet the right endpoint is a maxima. I guess this is an error in the textbook, but I just wanted to double check since my professor's office hours are not until the day after tomorrow.

4. ## Re: Why is this not a local maximum?

Anyone (any book) is free to define anything they want in any way they want. But 100 out of 100 mathematicians will say that x=a corresponds to a local maximum for that function you sketched.

That's assuming, of course, that a is actually in the domain! - meaning that the domain is like [a, b) and not like (a, b). Could *that* be the issue here?

f(x) = 3x on [4, 10) has a minimum of 12 at x=4, but no maximum. It "wants" to have a maximum of 30 at x=10, but x=10 isn't in the domain.

5. ## Re: Why is this not a local maximum?

Well, that was the graph they gave, without an interval. It would be ridiculous of the author to expect readers to assume that the interval is (a,b] just based on this graph, in my opinion at least. I'm just going to go with the textbook having an error.

6. ## Re: Why is this not a local maximum?

Based on your sketch, it is not a local maximum because a local maximum CANNOT be on an endpoint of an interval. Local maxima and minima must be on the interior of an interval. Make sense?

7. ## Re: Why is this not a local maximum?

alane - that's what his or her book says. That's what you say. I'd say otherwise, and Phizkid thinks otherwise. This is just a question of definitions. So I'm curious, is your statement here because you remember that that's what you were taught, or are you just accepting and reinforcing the book definition as Phizkid has stated it?

Wikipedia also says that only interior points are allowed to count as local maximums. So, maybe that is now the normative definition. Those authors are entitled to make whatever definitions they want - and I'm entitled to declare that their definitions are stupid.

8. ## Re: Why is this not a local maximum?

It is what I was taught, it is what about every math professor that I have talked to has said. I am going to side with the guy that has a PhD over you... sorry...

9. ## Re: Why is this not a local maximum?

1) You're presuming that I don't have a PhD.
2) "it is what about every math professor that I have talked to has said." Now, seriously, how many math professors have you actually asked about this? I've got big money saying the answer is zero.

10. ## Re: Why is this not a local maximum?

Actually, I have talked to 4 professors on this when I was learning it. Because I had the same question as PhizKid. They all said that it can't be a local maximum or minimum if it is an endpoint.

11. ## Re: Why is this not a local maximum?

You asked four different math PhD professors about exactly this question? It's a sure bet that you're lying. But - whatever...

12. ## Re: Why is this not a local maximum?

No 1 had a PhD... the rest just had masters. I should have specified.

13. ## Re: Why is this not a local maximum?

Originally Posted by alane1994
Actually, I have talked to 4 professors on this when I was learning it. Because I had the same question as PhizKid. They all said that it can't be a local maximum or minimum if it is an endpoint.
I have over thirty different calculus text books.
On this topic they are all over the place.
I like how Einar Hille solves the question. He simply says that $\displaystyle a$ is a endpoint maximum.

14. ## Re: Why is this not a local maximum?

I like that. It makes sense.