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I wonder how I can determine the minimum of the product between variables $x$ and $y$ (in terms of $\theta$), given that both $x < 1 - \theta$ and $y < 1 - \theta$, and $x + y = 1$?

So far I can establish that both $\theta < x < 1 - \theta$ and $\theta < y < 1 - \theta$, and that $\theta < 1/2$. What can I say about $xy$ from these?

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Given $x+y=1$, $xy$ is minimized by taking $x$ as small as possible (or as large as possible), subject to whatever other constraints are imposed. In particular, if you could take $y=1-\theta$ and $x=\theta$, so $xy=\theta(1-\theta)$, that would be a minimum. Since you have $y\lt1-\theta$, the minimum cannot be achieved - there is no minimum. You can get arbitrarily close to $\theta(1-\theta)$ by taking $y$ close to $1-\theta$.

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thanks. could you elaborate a bit more on why $\theta*(1-\theta)$ would be a minimum if $y=1-\theta$ and $x=\theta$? –  skyork Jun 21 '11 at 0:57
    
@skyork, let $x=(1/2)-r$, $y=(1/2)+r$ for some $r$, then $xy=(1/4)-r^2$, so it's clear that the bigger $r$ is, the smaller $xy$ is. So also the smaller $x$ is, the smaller $xy$ is. But the smallest $x$ can be is, well, there is no smallest, but it can't be smaller than $\theta$ (because $y$ can't be bigger than $1-\theta$), so $xy$ can't be smaller than $\theta(1-\theta)$. –  Gerry Myerson Jun 21 '11 at 1:32
    
understood. thanks! –  skyork Jun 21 '11 at 10:31
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