Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

Verified step by step guidance

1. Let the two numbers be \(x\) and \(y\). According to the problem, their difference is 14, so we can write the equation: \(x - y = 14\).
2. Express one variable in terms of the other using the difference equation. For example, solve for \(x\): \(x = y + 14\).
3. Write the product of the two numbers as a function of one variable: \(P = x \times y = (y + 14) \times y = y^2 + 14y\).
4. To find the minimum product, take the derivative of \(P\) with respect to \(y\) and set it equal to zero to find critical points: \(\frac{dP}{dy} = 2y + 14 = 0\).
5. Solve the derivative equation for \(y\), then substitute back into \(x = y + 14\) to find the corresponding \(x\). Finally, calculate the product \(P = x \times y\) using these values to find the minimum product.

Key Concepts

Formulating Equations from Word Problems

This involves translating the given conditions into mathematical expressions. Here, the difference between two numbers is fixed, so if one number is x, the other can be expressed as x minus 14. Setting up these relationships correctly is essential to analyze the problem.

Quadratic Functions and Their Properties

The product of the two numbers forms a quadratic expression in terms of one variable. Understanding how to write and manipulate quadratic functions allows us to analyze their behavior, such as finding minimum or maximum values.

Finding the Minimum Value of a Quadratic Function

A quadratic function with a positive leading coefficient opens upward and has a minimum point at its vertex. Using the vertex formula or completing the square helps find the input value that minimizes the product, which answers the problem.