1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculus, ordered set

  1. Sep 4, 2008 #1
    1. The problem statement, all variables and given/known data
    a,b,c,d in a ordered set K and b and d>0
    Show that [tex]\frac{a+c}{b+d}[/tex] stay between the minimum and max from [tex]\frac {a}{b}[/tex] and [tex]\frac {c}{d}[/tex]. Generalize for [tex]a_1,\hdots,a_n,b_1,\hdots,b_n \in K[/tex] with [tex]b_1\hdots,b_n >0[/tex] so [tex]\frac{a_1+\hdots+a_n}{b_1+\hdots+b_n}[/tex] is between the max and min elements from [tex]\frac{a_1}{b_1},\hdots,\frac{a_n}{b_n}[/tex]

    I could do it for the the first case but in a way it's impossible to generalize
    any ideas?
    tks in advance
    Last edited: Sep 4, 2008
  2. jcsd
  3. Sep 5, 2008 #2
    Re: Calculus, field set

    to correct, it`s a ordered field
  4. Sep 15, 2008 #3
    if u consider a/b<c/d
    u can do a/b - (a+c)/(b+d)}=(ad-bc)/(b(b+d))>0
    and c/d - (a+c)/(b+d)}=(bc-ad)/(b(b+d))>0
    and done

    but them using samething for generalizing i couldn't make it :(
    Last edited: Sep 15, 2008
  5. Sep 16, 2008 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Looks like proof by induction would work.
  6. Sep 16, 2008 #5


    User Avatar
    Science Advisor
    Homework Helper

    Let's redo the n=2 case in a way it will be easier to generalize. Write (a1+a2)/(b1+b2)=(b1/(b1+b2))*(a1/b1)+(b2/(b1+b2))*(a2/b2). Notice that the bi/(b1+b2) terms are positive and sum to 1. (This means (a1+a2)/(b1+b2) is in the 'convex hull' of the bi/ai.) If I replace the ai/bi by their minimum and maximum, what do I conclude?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Calculus, ordered set
  1. Order of Operations (Replies: 4)

  2. Calculus Intro (Replies: 3)

  3. Eigenvector order (Replies: 1)