Problem of Tartaglia (1500-1557): among all positive numbers a, b whose sum is 8, find those for which the product of the two numbers and their difference is largest.

Question

Problem of Tartaglia (1500-1557): among all positive numbers a, b whose sum is 8, find those for which the product of the two numbers and their difference is largest. (Hint: Let x = a - b and express abx in terms of x alone.)

a = ?

b = ?

Answer 1

a + b = 8

b = 8 - a

ab(a - b)

a(8 - a)(a - (8 - a))

a(8 - a)(2a - 8)

Expand

Differentiate

Set equal to 0