Question

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Answer 1

Let one number be x. Then, the other number is (16 − x). Let the sum of the cubes of these numbers be denoted by S(x). Then,

∴ By second derivative test, x = 8 is the point of local minima of S.
Hence, the sum of the cubes of the numbers is the minimum when the numbers are 8 and 16 − 8 = 8.