The product of two positive real numbers x and y is 30. Find the minimum possible value of their sum.

Answer 1

The product of two positive real numbers x and y is 30.

x*y = 30

Find the minimum possible value of their sum.

Sum = S = x + y

*solve the first equation for either x or y then substitute into the second equation

y = 30 / x

S = x + (30/x)

*Derive S, then set S' equal to zero.

S' = 1 - 30/x^2

0 = 1 - 30/x^2

*Solve for x

30/x^2=1

30=x^2

x=√30

*recall

y=30/x

y=30/√30=√30

*Therefore

The minimum value of the sum is

Sum = √30+√30 = 2√30

Answer 2

Since x and y are both positive, their square roots will be real. So √x - √y will also be real, and therefore

(√x - √y)² ≥ 0. But (√x - √y)² = x + y - 2 √(xy), and by assumption xy = 30. Thus x + y ≥ 2 √30.