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