Find two positive numbers whose product is 81 and whose sum is a minimum.

Ab=81, a=81/b

s=a+b using a from above in this we get:

s (b) = 81/b + b

s (b) = (81+b^2) / b

ds/db = (2b*b-81-b^2) / b^2

ds/db = (2b^2-81-b^2) / b^2

ds/db = (b^2-81) / b^2

d2s/db2 = (2b^3-2b^3+81) / b^4

d2s/db2=81/b^4 since b is positive we know that the acceleration is positive so that when ds/db=0 it is a minimum for s (b)

ds/db=0 only when b^2-81=0, b^2=81, b=9

The two positive numbers are 9 and 9.