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I Optimization problems

  1. Aug 11, 2017 #1
    I am little confused when it comes to optimization problems. For example, say we are given that ##x+y=2##, and are asked to maximize ##xy##. By AM-GM, we have that ##xy \le 1##. But why should this indicate that ##1## is the maximum value? Isn't it an equally true statement to claim that ##xy \le 2##, since the former interval is contained in the latter?
     
  2. jcsd
  3. Aug 11, 2017 #2

    andrewkirk

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    It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when ##x=y##. Arithmetic and Geometric Means are identical when all data are the same.
     
  4. Aug 11, 2017 #3
    I think it was the distinction between upper bound and least upper bound that I was looking for.
     
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