Problem 337 - Calculus Volume 1

Question

For the following exercises, consider two nonnegative numbers \(x\) and \(y\) such that \(x+y=10\) . Maximize and minimize the quantities. $$x y$$

Step-by-Step Solution

Answer

Max: 25 at \(x = 5, y = 5\); Min: 0 at \(x = 0, y = 10\) or \(x = 10, y = 0\).

Step 1: Understanding the Constraint

The problem states that we have two nonnegative numbers, \(x\) and \(y\), such that their sum is 10. This gives us the equation \(x + y = 10\).

Step 2: Express One Variable in Terms of the Other

From the equation \(x + y = 10\), we can express \(y\) in terms of \(x\): \(y = 10 - x\).

Step 3: Define the Product to Maximize or Minimize

The quantity we want to maximize or minimize is the product \(xy\). By substituting \(y = 10 - x\) into the equation for the product, we get: \(xy = x(10 - x) = 10x - x^2\).

Step 4: Find the Derivative of the Product Function

To find the maximum or minimum of the function, we need to take the derivative of \(f(x) = 10x - x^2\) with respect to \(x\). This gives us \(f'(x) = 10 - 2x\).

Step 5: Solve for Critical Points

Set the derivative equal to zero to find the critical points: \(10 - 2x = 0\). Solving for \(x\) gives \(x = 5\).

Step 6: Evaluate at Critical Point and Endpoints

Evaluate the product at \(x = 5\) and endpoints of the interval. At \(x = 5\), \(y = 10 - 5 = 5\) so \(xy = 5 \times 5 = 25\). At \(x = 0\), \(y = 10\) so \(xy = 0\). At \(x = 10\), \(y = 0\) so \(xy = 0\).

Step 7: Determine Maximum and Minimum Values

From the evaluations, the maximum product is 25 occurring at \(x = 5\) and \(y = 5\). The minimum product is 0 occurring at the endpoints \(x = 0, y = 10\) or \(x = 10, y = 0\).

Key Concepts

Critical Points

Critical points are essential when solving optimization problems. A critical point occurs where the derivative of a function is either zero or undefined. These points can indicate potential maxima, minima, or saddle points.

In this example, given the function that represents the product, \( f(x) = 10x - x^2 \), we found the derivative \( f'(x) = 10 - 2x \). By setting this derivative equal to zero, \( 10 - 2x = 0 \), we solve for \( x \) to find our critical point. This gives us \( x = 5 \).

• Critical points do not always ensure maxima or minima.
• It's necessary to evaluate the function to determine whether the critical point is a maximum, minimum, or a saddle point.

Once critical points are identified, checking the endpoints will confirm the function behavior across the interval. For our problem, evaluating the function at critical and endpoint values allowed us to determine that \( x = 5 \) implies a maximum value for the product.

Derivative

The derivative of a function is a major concept in calculus, representing the rate of change or slope of the function at any given point. Derivatives are fundamental when identifying critical points and understanding the behavior of functions.

In the problem, we find the derivative of \( f(x) = 10x - x^2 \). This yields \( f'(x) = 10 - 2x \), which is a linear function showing how the product changes as \( x \) varies.

• Setting the derivative to zero, \( 10 - 2x = 0 \), helps locate the critical points.
• The derivative also helps analyze the trend: if \( f'(x) \) changes from positive to negative, a maximum is found, and vice-versa for a minimum.

By calculating the derivative and solving where it equals zero, we uncover points that potentially maximize or minimize the function. This procedural step is crucial in determining the nature of these critical points, essential to optimization.

Maxima and Minima

Maxima and minima are the largest and smallest values, respectively, that a function can take in a given interval. In optimization problems, they represent the goals of either finding the highest or lowest output for a particular function.

In our exercise, we are tasked with finding both the maximum and minimum product of two nonnegative numbers where their sum is fixed at 10. By evaluating the function \( xy = 10x - x^2 \) at critical points and endpoints:

• At \( x = 5 \), \( y = 5 \), resulting in a product of \( 25 \), gives the maxima.
• At endpoints \( x = 0 \) or \( x = 10 \), yielding products of \( 0 \), indicates the minima.

Understanding where and why maxima and minima occur allows a comprehensive approach to solve optimization. Such insights reveal the function's behavior across the domain and clarify which values to pursue for optimal performance.

Nonnegative Numbers

Nonnegative numbers are numbers that are either positive or zero. They form a fundamental part of constraints in many mathematical problems, ensuring that solutions stay within permissible bounds.

In this problem, both \( x \) and \( y \) are nonnegative numbers, leading to specific conditions:

• \( x + y = 10 \) must always hold true.
• The solutions include endpoints where \( x = 0, y = 10 \) or \( x = 10, y = 0 \).

These conditions ensure that values stay realistic, as negative numbers for product optimization in this context don't make sense. By focusing only on nonnegative solutions, the problem caters to a practical situation where both numbers, \( x \) and \( y \), are aligned with real-world constraints. Speaking to this constraint is crucial in the problem’s framework and answers.