The product of two positive numbers is 16. Find the numbers if their sum is least and if the sum of one and the square of the other is least.

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The product of the two positive numbers is 16. Let one of the numbers be X, the other number is `16/X` .

The sum of the two numbers is `S = X + 16/X` . The value of S is  minimized for X such that `(dS)/(dX) = 0`

=> `1 - 16/X^2 = 0`

=> `X^2 = 16`

=> X = 4

16/X = 4

The sum of one of them and the square of the other is `S2 = X^2 + 16/X` . S2 is minimized for the value of X where `(dS2)/(dX) = 0`

=> `2X - 16/X^2 = 0`

=> `X^3 = 8`

=> X = 2

16/X = 8

The sum is the least for the set {4, 4} and the sum of one of the numbers and the square of the other is least for {2, 8}

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factors of 16:  1&16, 2&8, 4&4.  `1+16=17` , `2+8=10` , `4+4=8` .  For (a) the numbers are 4 and 4. 

`1^2 +16=17` `16^2+1=257 `

`2^2 +8=12`` 8^2 +2 = 66`

`4^2 + 4=20 `

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