Optimization problems

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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?
 

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  • #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.
 
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  • #3
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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.
I think it was the distinction between upper bound and least upper bound that I was looking for.
 

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