Question

Find two positive numbers satisfying the given requirements. The product is 216 and the sum is a minimum. (No Response) (smaller value) (No Response) (larger value)

Answer 1

`x1=√(216) \ and \ y1=\ √(216)`

Explanation:

Let the first number is x1 and other number is y1 then

`x1 * y1 =216`

Therefore

`y1=`
`(216)/(x1)`

also there sum is

`s1 =x1+y1`....Eq(1)

Putting the value of y1 in the previous equation

`s1\ =x1 + (216)/(x1)`........Eq(2)

Differentiate the the Eq(2) with respect to x1 we get

`(ds1)/(dx1) \ =\ 1+216*(1)/(-x1^(2) )`

`(ds1)/(dx1) \ =\ 1-(216)/(x1^(2) )\ =\ 0`

`{x1^(2) }\ =\ 216\\ x1=√(216)`

Putting the value of X1 in Eq(1) we get

`y1=(216)/(√(216) ) \\y1=(216*√(216) )/(216) \\y1=\ √(216)`

So
`x1=√(216) \ and \ y1=\ √(216)`