Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. In this context, the product of two numbers can be represented as a quadratic function, where the sum of the numbers is a constant. The maximum value of a quadratic function occurs at its vertex, which can be found using the formula x = -b/(2a).

Optimization

Optimization involves finding the maximum or minimum values of a function within a given set of constraints. In this problem, we are tasked with maximizing the product of two numbers while keeping their sum constant at 16. This requires applying techniques such as completing the square or using derivatives to identify the optimal solution.

Symmetry in Numbers

The concept of symmetry in numbers suggests that for a fixed sum, the product of two numbers is maximized when the numbers are equal. In this case, if the two numbers add up to 16, the maximum product occurs when both numbers are 8. This principle can be derived from the properties of arithmetic and geometric means, which state that the geometric mean is maximized when the numbers are equal.