What is the largest possible product for 2 even integers whose sum is 34 ?

F. 64 G. 68 H. 120 J. 240 K. 288

The correct answer is K.

Explanation: If n is one integer, then 34 - n is the other. To maximize n(34 - n), consider y = x(34 - x) = 34x - x 2 = -(x 2 - 34x) = -(x 2 - 34x + 289) + 289 = -(x - 17) 2 + 289.

The graph is a parabola that turns downward and has maximum point (17,289). Applying this to the task of maximizing n(34 - n) for even integers n, the closest even value to 17 is 16 (or 18), so n = 16 and 34 - n = 18 (or n = 18 and 34 - n = 16). The product is 288. So the even integers are 16 and 18. Their sum is 34, and their product is maximum.