Question

Two positive numbers have the property that their product is 2 and their sum is as small as possible. Find their sum.

Answer 1

a b = 2 ... a = 2 / b

s = a + b = 2/b + b

ds = db (-2/b² + 1)

ds/db = -2/b² + 1 = 0

b = √2

s = 2√2

Answer 2

Let one of them be x, and the other one y

xy = 2
y = 2/x

sum = x+y = x + 2/x
d(sum)/dx = 1 - 2/x^2 = 0 for a min sum

1 = 2/x^2
x^2 = 2
x = √2 , then y = 2/√2 = √2

their minimum sum = x+y
= √2 + √2 = 2√2

Answer 3

as usual, for a given perimeter (sum), a square has maximum area (product).