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Cauchy theorim question

  1. Jan 20, 2009 #1
    i got this question
    http://img412.imageshack.us/img412/3713/88436110xw9.gif [Broken]

    here is the solution:
    http://img297.imageshack.us/img297/6717/14191543qm1.th.gif [Broken]
    they are taking the minimal value
    and the maximal value
    the innequalitty that the write is correct min< <max

    but why??
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Jan 20, 2009 #2


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    Well, if you have
    [tex]f(x_1) + f(x_2) + \cdots + f(x_n)[/tex]
    and you know that each of the [itex]f(x_i)[/itex] is not greater than M, then you can write
    [tex]f(x_1) + f(x_2) + \cdots + f(x_n) \le M + M + \cdots + M = n \cdot M;[/tex]
    similarly for the minimum.

    It's simply applying the inequality that
    x + y <= M + y
    if x <= M.
  4. Jan 20, 2009 #3
    i agree with you
    but why they do that
    how is it linked to cauchy theorem
  5. Jan 20, 2009 #4


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    I don't know what it has to do with Cauchy's theorem, but it does have to do with the intermediate value theorem: for any value c between m and M (assuming some conditions on f which you didn't state) there is an x such that f(x) = c.
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