Results 1 to 14 of 14
Like Tree3Thanks
  • 2 Post By johnsomeone
  • 1 Post By Plato

Math Help - Why is this not a local maximum?

  1. #1
    Junior Member
    Joined
    Jul 2012
    From
    NY
    Posts
    40

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    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)."
    Last edited by johnsomeone; October 17th 2012 at 11:09 AM.
    Thanks from PhizKid and topsquark
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2012
    From
    NY
    Posts
    40

    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jul 2012
    From
    NY
    Posts
    40

    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.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member alane1994's Avatar
    Joined
    Oct 2012
    From
    Iowa, United States
    Posts
    65
    Thanks
    8

    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?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    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.
    Last edited by johnsomeone; October 17th 2012 at 01:15 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member alane1994's Avatar
    Joined
    Oct 2012
    From
    Iowa, United States
    Posts
    65
    Thanks
    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...
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    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.
    Last edited by johnsomeone; October 17th 2012 at 01:37 PM.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Junior Member alane1994's Avatar
    Joined
    Oct 2012
    From
    Iowa, United States
    Posts
    65
    Thanks
    8

    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.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    146

    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...
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Junior Member alane1994's Avatar
    Joined
    Oct 2012
    From
    Iowa, United States
    Posts
    65
    Thanks
    8

    Re: Why is this not a local maximum?

    No 1 had a PhD... the rest just had masters. I should have specified.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1

    Re: Why is this not a local maximum?

    Quote Originally Posted by alane1994 View Post
    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 a is a endpoint maximum.
    Thanks from PhizKid
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Junior Member alane1994's Avatar
    Joined
    Oct 2012
    From
    Iowa, United States
    Posts
    65
    Thanks
    8

    Re: Why is this not a local maximum?

    I like that. It makes sense.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Local Maximum and Minimum
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 5th 2011, 12:37 AM
  2. Replies: 4
    Last Post: March 21st 2011, 01:23 AM
  3. Replies: 6
    Last Post: January 5th 2011, 02:34 AM
  4. Local Maximum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 4th 2008, 07:49 PM
  5. local maximum problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 1st 2008, 12:23 PM

Search Tags


/mathhelpforum @mathhelpforum