Two positive numbers add up to 36. What is the largest possible value of their product?

Explanation

1. 01

   Let the numbers be $x$ and $36 - x$; the product is $P = 36x - x^2$.

2. 02

   $P' = 36 - 2x = 0$ at $x = 18$, and $P'' = -2 < 0$: a maximum, $18 \times 18 = 324$.