Find two integers whose sum is 12 and whose product is maximum. Solution We\'ll note the integers as x and y. The sum of the integers is 12. x + y = 12 y = 12 - x We also know that the product of integers is a maximum. We\'ll write the product of integers as: P = x*y We\'ll substitute y by (12-x) and we\'ll create the function p(x): p(x) = x*(12-x) We\'ll remove the brackets and we\'ll get: p(x) = 12x - x^2 The function p(x) is a maximum when x is critical, that means that p\'(x) = 0 We\'ll calculate the first derivative for p(x): p\'(x) = (12x - x^2)\' p\'(x) = 12 - 2x p\'(x) = 0 12 - 2x = 0 We\'ll divide by 2: 6 - x = 0 We\'ll subtract 6 both sides: -x = -6 We\'ll divide by -1: x = 6 So, x is the critical value and the integers are x = 6 and y = 6..