Two positive numbers have product 64. What is the least possible value of their sum?

Explanation

1. 01

   With the numbers $x$ and $\frac{64}{x}$, the sum is $S = x + \frac{64}{x}$, and $S' = 1 - \frac{64}{x^{2}} = 0$ at $x = 8$.

2. 02

   $S'' = \frac{128}{x^{3}} > 0$: a minimum, with both numbers 8 and sum 16.