Problem 17 - The Calculus with Analytic Geometry

Question

Find three numbers whose sum is \(N(N>0)\) such that their product is as great as possible.

Step-by-Step Solution

Answer

The three numbers are \( \frac{N}{3} \), yielding a maximum product of \( \frac{N^3}{27} \).

Step 1 - Understand the Problem

We need to find three numbers that add up to a given positive number, denoted by \( N \), such that their product is maximized.

Step 2 - Define Variables

Let the three numbers be denoted by \( x, y, \) and \( z \). We are given that \( x + y + z = N \) and we need to maximize the product \( P = xyz \).

Step 3 - Use Symmetry and Equal Distribution

Intuitively, the product of three numbers with a fixed sum is maximized when the numbers are as equal as possible. Therefore, let's consider the case where \( x = y = z \). Since \( x + y + z = N \), if the numbers are equal, then \( 3x = N \) yielding \( x = \frac{N}{3} \).

Step 4 - Calculate the Product

Substituting \( x = \frac{N}{3} \) into the product, we get: \[P = x^3 = \left( \frac{N}{3} \right)^3\]

Step 5 - Evaluate the Expression

Finally, the maximum product can be written as: \[P = \left( \frac{N}{3} \right)^3 = \frac{N^3}{27}\]

Key Concepts

calculus optimization

Maximizing a product using calculus involves finding the maximum value of an expression. To do this, you typically take the derivative and solve for critical points. In our exercise, we are asked to maximize the product of three numbers that sum up to a given number, N. The method involves:

• Setting up the equations and defining the variables (here, x, y, z).
• Expressing the product P in terms of a single variable.
• Using calculus to take the derivative of P.

In more detail: Given that x + y + z = N, we need to express P = xyz. To do this in the context of derivatives, you might set up a Lagrange multiplier, which is a method in calculus used to find the local maxima and minima of a function subject to equality constraints. But in this case, a simpler and more effective method is using symmetry and equal distribution.

symmetry in mathematics

Symmetry provides a powerful tool to simplify problems. In this exercise, symmetry suggests that the numbers x, y, and z should be as equal as possible to maximize their product when their sum is fixed at N. This is because products of numbers tend to be larger when the numbers are evenly distributed. For example:

• Consider two cases: splitting 10 into 5 and 5 (product 25) vs. splitting 10 into 6 and 4 (product 24). The first case gives a higher product due to equal distribution.

Similarly, if the three numbers sum up to N, they should be split equally as N/3 to maximize their product. The idea here leverages the balance and symmetry in mathematics to simplify and solve the problem efficiently.

equal distribution in algebra

Equal distribution in algebra deals with dividing quantities evenly to achieve optimal results. In the given problem, to maximize the product of three numbers with a sum N, equal distribution tells us that each number should be N/3. This is because:

• Products of equal numbers yield a higher product compared to unequal distributions.
• This concept can be visualized with simpler numbers or by using the AM-GM inequality (Arithmetic Mean-Geometric Mean inequality) which states that for non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean.

Applying this to our exercise, setting x, y, z to N/3 distributes N equally among the three numbers, leading to the maximal product.

product maximization

Product maximization is a classical problem in both algebra and calculus. The objective is to find the maximum value of a product given certain constraints. In this exercise, starting from x + y + z = N, the product P = xyz must be maximized. By setting x = y = z, we get:

• N = 3x, leading to x = N/3.
• Thus, the product P becomes \( P = (\frac{N}{3})^3 = \frac{N^3}{27} \).

This formula provides the maximum product of three numbers summing to N. The clear takeaway is that equal distribution simplifies and optimizes the product, ensuring it is as large as possible.