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