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.
