Find Two Positive Real Numbers Whose Product Is A Maximum

Finding Two Positive Real Numbers Whose Product is a Maximum: A Comprehensive Guide

Finding two positive real numbers whose product is a maximum is a classic optimization problem with applications across various fields, from economics to engineering. This seemingly simple problem introduces fundamental concepts in calculus and provides a powerful illustration of how mathematical tools can solve real-world challenges. This article will delve into the problem, exploring different approaches, providing detailed explanations, and addressing common questions. We'll examine both algebraic and calculus-based solutions, ensuring a comprehensive understanding for readers of various mathematical backgrounds.

Introduction: Understanding the Problem

The core problem is this: given a fixed sum of two positive real numbers, find the two numbers that yield the largest possible product. This isn't just an abstract mathematical exercise. Imagine a farmer with a limited amount of fencing wanting to maximize the area of a rectangular enclosure. The length and width of the rectangle represent the two numbers, their sum is the total fencing available, and their product represents the enclosed area. Optimizing this product is crucial for maximizing efficiency.

We can represent the problem mathematically. Let the two positive real numbers be x and y. Let their sum be a constant, S, so x + y = S. Our goal is to maximize the product P = xy.

Method 1: Algebraic Approach using Completing the Square

This method utilizes algebraic manipulation to reach the solution without explicitly using calculus. We start with the constraint x + y = S, which we can rewrite as y = S - x. Substituting this into the product equation, we get:

P = x(S - x) = Sx - x²

This is a quadratic equation representing a parabola that opens downward (since the coefficient of x² is negative). The maximum value of this parabola occurs at its vertex. We can find the x-coordinate of the vertex using the formula for the x-coordinate of the vertex of a parabola in the form ax² + bx + c, which is -b/2a. In our case, a = -1 and b = S. Therefore:

x = -S / (2 * -1) = S/2

Substituting this value of x back into y = S - x, we find:

y = S - S/2 = S/2

Therefore, the maximum product occurs when x = y = S/2. The maximum product itself is:

P_max = (S/2)(S/2) = S²/4

This algebraic method clearly demonstrates that the product is maximized when the two numbers are equal and each is half of the total sum.

Method 2: Calculus-Based Approach using Derivatives

Calculus provides a more general and powerful approach to solving optimization problems. We'll use the method of Lagrange multipliers, a technique well-suited for constrained optimization problems. However, for simplicity, and because the constraint is easily solvable for y, we can use a more direct approach involving derivatives.

We start with the equation P = xy, where y = S - x. Substituting, we get:

P(x) = x(S - x) = Sx - x²

To find the maximum, we take the derivative of P(x) with respect to x and set it to zero:

dP/dx = S - 2x

Setting the derivative to zero gives:

S - 2x = 0 => x = S/2

To confirm this is a maximum, we can take the second derivative:

d²P/dx² = -2

Since the second derivative is negative, this confirms that x = S/2 corresponds to a maximum. Substituting this value back into y = S - x, we get y = S/2, leading to the same conclusion as the algebraic method.

Method 3: AM-GM Inequality (Arithmetic Mean-Geometric Mean Inequality)

The Arithmetic Mean-Geometric Mean (AM-GM) inequality provides an elegant and concise solution. The AM-GM inequality states that for non-negative real numbers a and b:

(a + b)/2 ≥ √(ab)

Equality holds if and only if a = b. In our case, a = x and b = y, and we know x + y = S. Applying the AM-GM inequality:

(x + y)/2 ≥ √(xy)

Substituting x + y = S:

S/2 ≥ √(xy)

Squaring both sides:

S²/4 ≥ xy

Equality holds when x = y. Therefore, the maximum product, xy, is S²/4, which occurs when x = y = S/2.

Geometric Interpretation

The problem can be visualized geometrically. Consider the equation x + y = S. This represents a straight line with intercepts (S, 0) and (0, S) on the coordinate plane. The product xy can be interpreted as the area of a rectangle with sides x and y. The line x + y = S represents all possible combinations of x and y that sum to S. The rectangle with the largest area is a square with side length S/2, resulting in an area of S²/4. This reinforces the result that the maximum product occurs when x = y = S/2.

Illustrative Examples

Let's consider some numerical examples:

• Example 1: If the sum of two numbers is 10 (S = 10), then the two numbers that maximize the product are x = 5 and y = 5, giving a maximum product of 25.

• Example 2: If the sum is 100 (S = 100), the numbers are x = 50 and y = 50, with a maximum product of 2500.

• Example 3: Suppose a farmer has 100 meters of fencing. To maximize the area of a rectangular enclosure, they should create a square with sides of 25 meters each, resulting in an area of 625 square meters.

Frequently Asked Questions (FAQ)

• Q: What if the numbers are not restricted to be positive? If negative numbers are allowed, the product can be arbitrarily large (or small). There would be no maximum.

• Q: Can this be extended to more than two numbers? Yes. The AM-GM inequality can be generalized to n non-negative real numbers. For example, for three numbers whose sum is S, the maximum product occurs when all three numbers are equal to S/3.

• Q: What if the relationship between the numbers isn't a simple sum? If the relationship is different, you would need to use a different optimization technique, likely involving Lagrange multipliers if the relationship forms a constraint.

• Q: Are there any real-world applications beyond the fencing example? Many! This principle applies in diverse areas, including:

  • Business: Determining optimal pricing strategies to maximize revenue.
  • Engineering: Designing structures with maximum strength or efficiency given material constraints.
  • Finance: Portfolio optimization to maximize returns given risk constraints.

Conclusion: The Power of Optimization

Finding two positive real numbers whose product is a maximum, while seemingly simple, unveils fundamental concepts in mathematics and optimization. The problem elegantly demonstrates the power of different approaches – algebraic manipulation, calculus, and the AM-GM inequality – to achieve the same solution. Understanding this problem provides a strong foundation for tackling more complex optimization problems in various fields, highlighting the practical applications of mathematical theory in everyday scenarios. The key takeaway is that for a fixed sum, the maximum product is achieved when the numbers are equal. This principle has far-reaching implications in diverse areas, illustrating the practicality and beauty of mathematical optimization.