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Min clearly means minimum and max maximum, but I am confused about a question that says "With $x, y, z$ being positive numbers, let $xyz=1$, use the AM-GM inequality to show that min max $[x+y,$ $x+z,$ $y+z]=2$ What does this mean? (I am not looking for the answer this particular question, but just what "min max" means.

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    $\begingroup$ The minimum value of the maximum: Prove that out of the three numbers $x+y$, $x+z$ and $y+z$, the largest of the three is at least $2$. $\endgroup$ – ShreevatsaR Jun 19 '13 at 3:49
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The meaning will depend on context. Here it means that for each triple $\langle x,y,z\rangle$ such that $xyz=1$ we find the maximum of $x+y,x+z$, and $y+z$, and then we find the smallest of those maxima: it’s

$$\min\Big\{\max\{x+y,x+z,y+z\}:xyz=1\Big\}\;.$$

In general it will be something similar: you’ll be finding the minimum of some set of maxima.

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