Question

Look for two positive numbers whose product is 36 and whose sum is a minimum.

Answer 1

We want to minimize s(x, y) = x + y such that xy = 36.

From the constratint we have y = 36/x, so the sum can also be written as

s(x, 36/x) = x + 36/x

Differentiate s with respect to x and find the critical points of s :

1 - 36/x² = 0   ⇒   x² = 36   ⇒   x = ±6

Both x and y must be positive, so it follows that x = y = 6, which gives a minimum sum of s(6, 6) = 12.