Find the Maximum Product by Sum Calculator (Two Integers)

Welcome to the free online calculator to calculate the maximum product based on the sum of two integers. For example, the sum of two numbers is 64. What is the maximum product of these two numbers?

There are two methods, one is the enumeration method, and the other is the derivative method.

Method 1: Enumeration Method

This is the most common method. Enumerating the various possibilities directly, then calculating each possible product, and finally determining which product is the largest.

For example, what is the maximum value of the product of two numbers whose sum is 10?

The pairs of integers that sum to 10 are as follows:

1 + 9 = 10

2 + 8 = 10

3 + 7 = 10

4 + 6 = 10

5 + 5 = 10

Their product is

1 * 9 = 9

2 * 8 = 16

3 * 7 = 21

4 * 6 = 24

5 * 5 = 25

So, the maximum product is 25.

This method is simple but has the disadvantage that if the sum of two numbers is large, there are many cases to enumerate.

Method 2: Derivative Method

Let the two numbers be x and y. The sum of them is s.

x + y = s

Let y be represented by x.

y = s – x

Thus, the product of these two numbers is

P = xy = x(s-x) = sx – x²

Now, according to the definition of the derivative, we want to calculate the maximum value of P, just take the derivative of sx- x² with respect to x and set it to 0.

d(sx – x²)dx = s – 2x = 0

s – 2x = 0

2x = s

x = s2

It can be seen that the number is equal to half of the sum, and their product is the largest.

Conclusion: if the sum is even, just divide by 2 and take the middle number. If the sum is odd, divide by 2 and take the two adjacent integers of the quotient.

In the above example: what is the maximum value of the product of two numbers whose sum is 10?

102 = 5

5 * 5 = 25

The maximum product of them is also 25.

Another example, the sum of two numbers is 15. What is the maximum product of them?

First, divide 15 by 2

152 = 7.5

Take two integers around 7.5 are 7 and 8.

So, the maximum product of these two numbers is

7 * 8 = 56

Use enumeration to verify.

1 + 14 = 15,  1 * 14 = 14

2 + 13 = 15,  2 * 13 = 26

3 + 12 = 15,  3 * 12 = 36

4 + 11 = 15,  4 * 11 = 44

5 + 10 = 15,  5 * 10 = 50

6 + 9 = 15,  6 * 9 = 54

7 + 8 = 15,  7 * 8 = 56

Again, the result is 56.

Therefore, the above conclusion is feasible.

Using the calculator is simple, follow these 2 steps:

Next, let’s look at some examples.