What is the Maximum Product of Two Numbers Whose Sum is 16?

The maximum product of two numbers whose sum is 16 is 64.

Understanding the Principle of Maximization

When the sum of two numbers is fixed, their product is maximized when the numbers are as close to each other as possible. In cases where the numbers can be equal, the maximum product occurs precisely when the two numbers are identical. This principle is a fundamental concept in algebra and optimization.

Mathematical Proof and Solution

To find the exact answer, we can use a straightforward algebraic approach:

1. Define the numbers: Let the two numbers be x and y.
2. State the given condition: Their sum is 16, so x + y = 16.
3. Express one number in terms of the other: From x + y = 16, we can write y = 16 - x.
4. Formulate the product function: We want to maximize the product P. So, P = x * y.
5. Substitute to get a single-variable function: Substitute y = 16 - x into the product equation:
   P(x) = x(16 - x)
P(x) = 16x - x^2

This equation P(x) = -x^2 + 16x represents a quadratic function, which graphs as a parabola opening downwards. The maximum value of such a function occurs at its vertex.

• Finding the vertex: For a quadratic function in the form ax^2 + bx + c, the x-coordinate of the vertex is given by the formula x = -b / (2a).
  • In P(x) = -x^2 + 16x, we have a = -1 and b = 16.
  • Therefore, x = -16 / (2 * -1) = -16 / -2 = 8.

6. Determine the two numbers:

   • When x = 8, the first number is 8.
   • The second number y = 16 - x = 16 - 8 = 8.
   • So, the two numbers are 8 and 8.

7. Calculate the maximum product:

   • The maximum product is 8 * 8 = 64.

This method clearly demonstrates that dividing the sum equally between the two numbers yields the greatest possible product. For more on maximizing quadratic functions, you can explore resources like Khan Academy's explanation on finding the maximum of a quadratic function.

Visualizing the Product

To further illustrate this, consider a few pairs of numbers that sum to 16 and their corresponding products:

First Number (x)	Second Number (16-x)	Product (x * (16-x))
1	15	15
3	13	39
5	11	55
7	9	63
8	8	64
9	7	63
11	5	55
15	1	15

As the table shows, the product increases as the numbers get closer to each other, reaching its peak when they are both 8.

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality

For non-negative numbers, this principle is formally expressed by the Arithmetic Mean-Geometric Mean (AM-GM) inequality. It states that for any two non-negative numbers a and b, the arithmetic mean is always greater than or equal to the geometric mean:

(a + b) / 2 >= sqrt(a * b)

Equality holds if and only if a = b.

In our case:

• a + b = 16
• (16) / 2 >= sqrt(a * b)
• 8 >= sqrt(a * b)
• Squaring both sides: 64 >= a * b

This shows that the maximum product a * b can be is 64, and this maximum is achieved when a = b, which means a = 8 and b = 8.

Practical Applications

This optimization principle extends to various real-world scenarios, such as:

• Maximizing Area: Given a fixed perimeter for a rectangular enclosure, a square (where length and width are equal) will always yield the maximum area.
• Engineering Design: Optimizing material usage for maximum output or efficiency.
• Economics: Modeling scenarios where resources need to be allocated to maximize profit or utility.

Understanding how to maximize a product given a fixed sum is a fundamental concept with broad utility.