Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook