Questions: Among all pairs of numbers whose sum is 8, find a pair whose product is as large as possible. What is the maximum product? The pair of numbers whose sum is 8 and whose product is as large as possible is . (Use a comma to separate answers.) The maximum product is .

Solution Steps

To solve this problem, we need to find two numbers, \( x \) and \( y \), such that their sum is 8 and their product is maximized. We can express one number in terms of the other (e.g., \( y = 8 - x \)) and then express the product as a function of a single variable. By finding the derivative of this function and setting it to zero, we can find the critical points that will help us determine the maximum product.

Let \( x \) and \( y \) be the two numbers such that \( x + y = 8 \). We can express \( y \) in terms of \( x \): \[ y = 8 - x \]

The product \( P \) of the two numbers can be expressed as: \[ P = x \cdot y = x(8 - x) = 8x - x^2 \]

To maximize the product, we take the derivative of \( P \) with respect to \( x \): \[ \frac{dP}{dx} = 8 - 2x \]

Setting the derivative equal to zero to find critical points: \[ 8 - 2x = 0 \implies 2x = 8 \implies x = 4 \]

Substituting \( x = 4 \) back into the expression for \( y \): \[ y = 8 - 4 = 4 \] Thus, the pair of numbers is \( (4, 4) \). The maximum product is: \[ P = 4 \cdot 4 = 16 \]

Final Answer