Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

Solution:

Consider x as the first term and y as the second term

x + 2y = 100

x = 100 - 2y

We know that

Product of two numbers is x × y = xy

(100 - 2y) × y

Let us consider this function as P(x)

P(x) = y(100 - 2y) = 100y - 2y²

In order to determine the maximum value

We first have to find derivative P’(x)

P’(x) = 100 - 4y

To find the critical number, let us set P’(x) = 0

At one of the critical numbers, the maximum value of P(x) will occur.

Here there is only one critical number.

100 - 4y = 0

4y = 100

Divide both sides by 4

y = 25

One number is 25, the other number is x = 100 - 2(25) = 100 - 50 = 50

Therefore, two positive numbers are 50 and 25.

Find two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum.

Summary:

The two positive numbers such that the sum of the first and twice the second is 100 and their product is a maximum are 50 and 25.