Find real positive numbers x,y that give the largest value of s = minim(x,y+(1/x),1/y)?

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Since the problem provides the information that s represents the minimum value between the numbers `x, y + 1/x, 1/y` , hence, you may consider the following inequalities, such that:

`{(s <= x),(s <= y + 1/x),(s <= 1/y):} => {(1/s >= x),(s <= y + 1/x),(1/s >= y):} => y + 1/x <= 1/s + 1/s = 2/s => s <= 2/s `

`s^2 <= 2 `

You need to solve the inequality `s^2 <= 2` for `x, y > 0, x, y in R` , such that:

`s <= sqrt2`

Since `x>= s` yields:

`{(x >= s),(s <= sqrt2):} => x = sqrt2`

Since `s <= 1/y` yields:

`{(s <= 1/y),(s <= sqrt2):} => 1/y = sqrt 2 => y = 1/sqrt2 => y = sqrt2/2`

**Hence, evaluating the real positive numbers `x,y` that give the largest value of `s` , under the given conditions, yields **`x = sqrt2, y = sqrt2/2.`

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