91Ó°ÊÓ

The arithmetic mean is the simple average of a set of numbers. It is calculated by adding the numbers together and dividing by the number of inputs. For our exercise, we have only two numbers, namely, \(x\) and \(y\), which sum up to a constant value, 40.
To apply the arithmetic mean, or AM, it is expressed as:

• Arithmetic Mean: \( \frac{x + y}{2} \)

Given the condition \(x + y = 40\), we can substitute and simplify the AM:

• \( \frac{x + (40-x)}{2} = 20 \)

This provides a vital understanding that, specifically for two positive values that sum up to 40, the arithmetic mean simply equals 20. This forms the baseline for comparing the arrangement or assignment of numbers in such optimization tasks.

The geometric mean is another type of average that is particularly useful when dealing with rates and proportional growth. For two numbers \(x\) and \(y\), the geometric mean is the square root of their product:

• Geometric Mean: \( \sqrt{x \cdot y} \)

In our problem, we apply the AM-GM inequality, which states that for any non-negative numbers \(x\) and \(y\), the arithmetic mean is always greater than or equal to the geometric mean:

• \( \frac{x+y}{2} \geq \sqrt{x \cdot y} \)
• With our inputs: \( 20 \geq \sqrt{x \cdot y} \)

This means that under our constant sum of 40, the greatest value of \(\sqrt{x \cdot y}\) is 20, leading us to the maximum product of \(x\) and \(y\). By maximizing the geometric mean, we set conditions optimal to achieve the highest product under given constraints.

Optimization deals with finding the best possible solution under given constraints. In this exercise, our task is to find the maximum product of \(x\) and \(y\) when both variables are positive and their sum equals 40. This calls for applying mathematical techniques to determine critical values and conditions.
Using the AM-GM inequality, we establish that equivalent distribution of \(x\) and \(y\) (i.e., making \(x = y\)) gives us the maximum product. Substituting \(x = 40 - x\), we solve:

• \(x = 20\)
• Thus, \(y = 20\)
• Maximum Product \(x \cdot y = 20 \cdot 20 = 400\)

By realizing these steps, we see that treating it as an optimization problem fits because we are effectively striving to achieve the best result within a boundary. Here, ensuring the most equal distribution of values under sum constraints was determined as optimal.