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Problem with supremum and infimum examples

  1. Oct 26, 2011 #1
    So I've got a calculus test in a week, and I'm studying for it but I can't understand some examples our professor has given us. So, he says:

    1) A = { x[itex]\in[/itex] ℝ: (x-a)(x-b)(x-c) < 0 } , a<b<c. Find the supA and infA.

    In the solution of his example he says. It is easy to see that A = (-∞,a)[itex]\cup[/itex](b,c) so infA = -∞ and supA = c. How did he find that A = (-∞,a)[itex]\cup[/itex](b,c) ?

    2) B = {1 +(-1)n:n[itex]\in[/itex] N}. Find the supB and infB.

    In the solution he says. Obviously B={0,1} which we can compute if we make n=2k (even) and n=2k+1 (odd)
    so infB=0 and supB=1. So if n=2k wouldn't B be 2? Is this a mistake my professor made?
     
  2. jcsd
  3. Oct 26, 2011 #2

    micromass

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    The easiest way to solve this is by making a sign diagram (or whatever you call it).
    Let me find out when (x-a)(x-b)<0.

    If x<a, then x-a<0. If x>a, then x-a>0. If x<b, then x-b<0, If x>b, then x-b>0. So putting these in a diagram yields

    ------------------ a +++++++++++++++++++++++
    -------------------------- b +++++++++++++++++

    Multiplying the two gives us

    ++++++++++++++ a ------ b +++++++++++++++++

    So we see that the function is negative between a and b. So A=(a,b) here.

    Now try to find it for three terms.




    Yes, it needs to be B={0,2}
     
  4. Oct 27, 2011 #3
    Thanks for clearing these things up.
     
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