The two positive numbers whose sum is 20 and whose product is the maximum are

Georgia State University

Cynthia R.

Algebra

8 months, 2 weeks ago

Let the two positive numbers be ‘x’ and ‘y’

Given product is 20

⇒ xy = 20

⇒ y = `20/x`

Sum S = x + y

S = `x + 20/x`

`"dS"/("d"x) = 1 - 20/x^2`

For maximum or minimum, `"dS"/("d"x)` = 0

x2 – 20 = 0

x2 = 20

x = `+  2sqrt(5)`

x = `- 2sqrt(5)` is not possible

`("d"^2"S")/("d"x^2) = 40/x^3`

At x = `2sqrt(5), ("d"^2"S")/("d"x^2) > 0`

∴ Sum ‘S’ is minimum when x = `2sqrt(5)`

y = `20/(2sqrt(5)) = 2sqrt(5)`

Minimum sum = `2sqrt(5) + 2sqrt(5)`

= `4sqrt(5)`