Let the two numbers be y and x, then
xy = 192
y = 192/x
dy/dx = -192/x^-2 . . . (1)
Sum (S) = y + 3x
If the sum is minimum, then
dS/dx = dy/dx + 3 = 0
dy/dx = -3
From (1), -192/x^2 = -3
3x^2 = 192
x^2 = 192/3 = 64
x = sqrt(64) = 8
y = 192/8 = 24
The two positive numbers are 8 and 24.
