Question

Find positive numbers x and y satisfying the equation xy=12 such that the sum 2x+y is as small as possible.

Answer 1

We are given equations as

$xy=12$

Firstly, we will solve for y

$y=\frac{12}{x}$

now, we are given sum as

$S=2x+y$

now, we can plug back y

$S=2x+\frac{12}{x}$

Since, we have to minimize it

so, we will find it's derivative

$S'=2-\frac{12}{x^2}$

now, we can set it to 0

and then we can solve for x

$S'=2-\frac{12}{x^2}=0$

$x=\sqrt{6},\:x=-\sqrt{6}$

Since , both numbers are positive

so, we will only consider positive value

$x=\sqrt{6}$

now, we can find y-value

$y=\frac{12}{\sqrt{6}}$

$y=2\sqrt{6}$

So, two positive numbers are

$x=\sqrt{6}$

$y=2\sqrt{6}$............Answer