Question

Find positive numbers x and y satisfying the equation xyequals15 such that the sum 3xplusy is as small as possible. Let S be the given sum. What is the objective function in terms of one​ number, x? Sequals nothing ​(Type an​ expression.) The interval of interest of the objective function is nothing. ​(Simplify your answer. Type your answer in interval​ notation.) The numbers are xequals nothing and yequals nothing. ​(Type exact​ answers, using radicals as​ needed.)

Answer 1

`x = √(5)\\\\y = (15)/( √(5) )`

Explanation:

According to the information of the problem

`xy = 15`

And

`S = 3x+y`

If you solve for
`y` on the first equation you get that

`y = {\displaystyle (15)/(x)}`

then you have that

`S = {\displaystyle 3x + (15)/(x) }`

If you find the derivative of the function you get that

`S' = {\displaystyle 3 - (15)/(x^2)} = 0\\`

The equation has two possible solutions but you are looking for POSITIVE numbers that make
`S` as small as possible.

Then

`x = √(5)\\\\y = (15)/( √(5) )`