Results 1 to 4 of 4

Math Help - Optimization

  1. #1
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823

    Optimization

    When working an optimization problem, when should I consider the domain of the primary equation?


    Also, when it is determined that the interval of which the variable is an element is open, do I consider the endpoints? For example. Let's say that f is continuous on the open interval (a,b) and has a relative maximum at  x=c. But at the point x=d, with a<d<b, f(d)>f(c). This is strange because, when I differentiate with respect to the variable and find all values for which f'(x)=0, all I get is the relative extrema, And I can't consider the endpoints because they are not in the domain. What do I do?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member apcalculus's Avatar
    Joined
    Apr 2009
    From
    Boston
    Posts
    293
    Quote Originally Posted by VonNemo19 View Post
    When working an optimization problem, when should I consider the domain of the primary equation?


    Also, when it is determined that the interval of which the variable is an element is open, do I consider the endpoints? For example. Let's say that f is continuous on the open interval (a,b) and has a relative maximum at  x=c. But at the point x=d, with a<d<b, f(d)>f(c). This is strange because, when I differentiate with respect to the variable and find all values for which f'(x)=0, all I get is the relative extrema, And I can't consider the endpoints because they are not in the domain. What do I do?
    If the domain is a closed interval you use the 'closed interval method', where you essentially list the y-values of all critical numbers and the y-values of the endpoints, then pick out the least or the greatest.

    I am not sure I completely understand the second part of your question. Remember that critical numbers are x values for which f'(x) is zero or undefined.

    My two cents.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Quote Originally Posted by apcalculus View Post
    I am not sure I completely understand the second part of your question.
    What I'm saying is that an open interval can contain a relative max, but not an absolute max. Think of the extreme value theorem, if continuity goes, then so does the theorem.

    So, again, when optimizing, if there be a relative maximum on (a,b) then there is some point, call it c, such that f'(c)=0. this is considered a relative maximum at x=c. But, consider what happens if there is another number (or an infinite set of numbers), call it d in (a,b) for which f(d)>f(c), you can see that the Optimization method has failed at locating the absolute maximum on (a,b). The problem is that the endpoints are non-inclusive.

    Do you see what I mean now. I tried to express myself as best I could.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member apcalculus's Avatar
    Joined
    Apr 2009
    From
    Boston
    Posts
    293
    Yes. You are correct. Absolute (or global) extrema need not exist when the function is only defined in an open interval. The closed interval with continuity guarantees the existence of both types of global extrema. (Extreme Value Theorem)

    f(x) = -x^2 + 1 on (-1, 0) union (0, 1)
    f(0) = 1/2

    ... is an example of a function with no types of global extrema.

    Good luck!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. optimization help!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 12th 2009, 12:54 AM
  2. Optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2009, 02:09 PM
  3. Optimization
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 29th 2009, 10:56 AM
  4. optimization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2008, 10:47 AM
  5. Optimization
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: October 13th 2008, 06:44 PM

Search Tags


/mathhelpforum @mathhelpforum