Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express a positive number \(N\) as a sum of three positive numbers such that the product of these three numbers is a maximum.

Step-by-Step Solution

Answer

Maximize the product by setting all three numbers equal: \(x = y = z = \frac{N}{3}\).

Step 1: Understand the Problem

We need to express a positive number \(N\) as a sum of three positive numbers \(x, y, z\) such that their product \(x\cdot y\cdot z\) is maximized. This involves using algebraic techniques to find the optimal values of \(x, y,\) and \(z\).

Step 2: Express Variables

Let the three numbers be \(x, y,\) and \(z\), such that \(x + y + z = N\). Our goal is to express the product \(P = x\cdot y\cdot z\) and determine conditions for its maximum.

Step 3: Use Symmetry to Simplify

Assume symmetry in the numbers for simplicity, as this often leads to maximization when dealing with products. Assign \(x = y = z\) and use the condition \(x + y + z = N\). We simplify to \(3x = N\), so \(x = \frac{N}{3}\).

Step 4: Find the Product in Terms of One Variable

Substituting \(x = \frac{N}{3}\) into the product equation gives us \(P = x\cdot x\cdot x = \left(\frac{N}{3}\right)^3\).

Step 5: Verify Constraints

Ensure all individual terms \(x\), \(y\), and \(z\) are positive. Since \(N\) is positive and \(x = \frac{N}{3}\), the individual terms \(x, y, z\) are positive.

Step 6: Conclusion on Maximizing the Product

The product \(x\cdot y\cdot z\) is maximized when all three numbers \(x = y = z = \frac{N}{3}\).

Key Concepts

Product of Numbers

In mathematics, the product of numbers refers to the result when two or more numbers are multiplied together. In the context of our exercise, the goal is to split a number into three parts whose product is as large as possible. This is important because it plays a key role in many optimization problems.
When maximizing the product of numbers, it is often effective to use symmetry. This means trying to make the numbers equal or close to each other as much as possible. The reason behind this is that, for a fixed sum, the product is maximized when the numbers are equal.
To illustrate, if you have to split 12 into three parts to maximize the product, using 4, 4, and 4 will yield a larger product than using 6, 5, and 1. This approach is not only elegant but also often leads to simpler calculations.

Sum of Numbers

The sum of numbers is the total obtained by adding two or more numbers together. In our maximization problem, the sum refers to the positive number \( N \) which is expressed as \( x + y + z \). This sum is crucial as it serves as a constraint for which other numbers are derived.
Thinking about the sum helps set the boundaries when dividing numbers. It tells us how to break down \( N \) into smaller parts that will subsequently be used to maximize the product.

• For instance, knowing \( x + y + z = N \) guides how we calculate each variable.
• It allows us to manipulate this equation to express variables simplistically, aiding in solving the problem using algebraic techniques.

Breaking down a number into a sum is a foundational step, serving as the blueprint for both the maximization process and verifying solutions.

Calculus

Calculus is a branch of mathematics dealing with change and motion, through derivatives and integrals. It is instrumental in optimization problems, where we find conditions for maxima or minima of functions.
In this exercise, though we did not explicitly use calculus, the concept helps to understand how functions behave:

• Derivatives can help identify maximum values. For a function like \( f(x) = x \cdot y \cdot z \), taking its derivative helps locate points where it might reach a peak.
• However, by recognizing the symmetry in this case, we simplified calculations by avoiding explicit use of calculus derivatives.

Even though calculus was bypassed directly here, it forms the theoretical backbone aiding in grasping why certain algebraic strategies, like symmetry, work effectively for such problems.

Algebraic Techniques

Algebraic techniques involve using operations and the properties of numbers to solve equations or optimize functions. In our problem, algebra is crucial in simplifying expressions and finding solutions.
The initial step is to represent the problem in terms of equations, like \( x + y + z = N \) and the product \( P = x \cdot y \cdot z \). Using algebra, we:

• Substituted variables (e.g., expressing \( x \) in terms of \( N \) as \( x = \frac{N}{3} \)) to simplify our calculations.
• Used algebraic properties such as symmetry to suggest that equal numbers produce maximum products.
• Tested conditions easily, as algebra provides a structured way to check whether variables meet required constraints, like positivity.

Mastering algebraic techniques empowers us to manipulate and play with equations, enabling an efficient pathway to solutions.