root(2)

First prove min(a, b + 1/a, 1/b) < =root(2)
If a <=root(2)  or b >= root(2)/2 =>done immediately.
Otherwise if a>=root(2) and b<=root(2)/2,
then b+1/a<= root(2)/2 + 1/root(2) = root(2).

Second min(a, b + 1/a, 1/b) =root(2) when
a=root(2) and b=root(2)/2.

Therefore max min(a, b + 1/a, 1/b) =root(2).




 

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