# The sum of two non negative numbers is 20. Find the numbers if the sum is as large as possible and find the numbers if one number plus the square root of this other is as small as possible.

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The sum of two non negative numbers is 20.

The sum of the numbers is 20, the first part of the question that is to find the numbers if the sum is as large as possible does not make sense.

The numbers for the second part that is to find the numbers if one number plus the square root of the other is as small as possible can be determined.

If one the numbers is taken as x. The second number is 20 - x. The required value that has to be minimized is `y = x + sqrt(20 - x) `

Solve y' = 0

`y' = 1 - 1/(2*sqrt(20 - x))`

`1 - 1/(2*sqrt(20 - x)) = 0`

=> `1 = 1/(2*sqrt(20 - x))`

=> `2*sqrt(20 - x) = 1`

=> `sqrt(20 - x) = 1/2`

=> `20 - x = 1/4`

=> x = 19.75

**The numbers for which one number plus the square root of the other is as small as possible are 19.75 and 0.25**