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#1 2006-03-15 07:44:50

razor
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minimum

We have the real  positive(only) numbers space of dimension two.
C={(x,y) belong to RxR and x+2*y<=2) and f(x,y)=min{x,y}.What is the minimum,x or y?

#2 2006-03-15 16:22:26

Ricky
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Re: minimum

If you're looking for a definite answer, there isn't one.

if x > 2/3, y < x
if y > 2/3, x < y

Any other restrictions lead to arbitrary (unknown) results.

x + 2y <= 2
y <= -x/2 + 1

The way I thought about it was to first assume that x=y.  Then 3y <= 2.   So y <= 2/3.  So if y increases any more, x must decrease.  Even if y increases by a small amount, x must decrease by any small amount, and thus, x < y.  The same is true for if x increases.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."