# 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. 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...

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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}

Approved by eNotes Editorial Team 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 `

Approved by eNotes Editorial Team