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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.)

Respuesta :

Answer:

[tex]x = \sqrt{5}\\\\y = \frac{15}{ \sqrt{5} }[/tex]

Step-by-step explanation:

According to the information of the problem

[tex]xy = 15[/tex]

And

[tex]S = 3x+y[/tex]

If you solve for [tex]y[/tex] on the first equation you get that

[tex]y = {\displaystyle \frac{15}{x}}[/tex]

then you have that

[tex]S = {\displaystyle 3x + \frac{15}{x} }[/tex]

If you find the derivative of the function you get that

[tex]S' = {\displaystyle 3 - \frac{15}{x^2}} = 0\\[/tex]

The equation has two possible solutions but you are looking for POSITIVE numbers that make [tex]S[/tex] as small as possible.

Then

[tex]x = \sqrt{5}\\\\y = \frac{15}{ \sqrt{5} }[/tex]