Problem 64 - Calculus Early Transcendentals

Question

Let \(R\) be a closed bounded set in \(\mathbb{R}^{2}\) and let \(f(x, y)=a x+b y+c,\) where \(a, b,\) and \(c\) are real numbers, with \(a\) and \(b\) not both zero. Give a geometrical argument explaining why the absolute maximum and minimum values of \(f\) over \(R\) occur on the boundaries of \(R\)

Step-by-Step Solution

Answer

In conclusion, we can show that the absolute maximum and minimum values of the linear function \(f(x, y) = ax + by + c\) will occur on the boundaries of the closed bounded set \(R\) in \(\mathbb{R}^{2}\), as the plane represented by the function keeps going up or down indefinitely in a particular direction. By considering the geometrical interpretation of the problem, we find that since the set \(R\) has a finite boundary, the highest and lowest points must necessarily be on the boundary of the set \(R\). Thus, for any given closed and bounded set \(R\), the absolute maximum and minimum values of the function must occur on the boundaries of the set \(R\).

Step 1: Define the problem

We have the function \(f(x, y) = ax + by + c\) and we need to show that the absolute maximum and minimum values occur on the boundaries of the closed bounded set \(R\) in \(\mathbb{R}^{2}\).

Step 2: Identify the geometric representation of the function

Since \(f(x, y)\) is a linear function, its graph is a plane in \(\mathbb{R}^{3}\). This plane can be written in the general form as \(z = ax + by + c\). The plane keeps going up or down indefinitely as it moves further in a particular direction.

Step 3: Identify the characteristics of the set R

The set R is given to be a closed and bounded set. This means that it is compact and contains all its limit points. It also means that the set R has an outer boundary.

Step 4: Geometrical interpretation

We need to find the highest and lowest points of the graph of \(f(x, y) = ax + by + c\) that intersect the set \(R\). Since the plane keeps going up or down, it will reach its maximum and minimum values at the boundaries of the closed bounded set \(R\). To better illustrate this, let's consider an example: Let's assume that \(R\) is a circle and \(f(x, y)\) is a plane passing through the circle. Now, if we start at any point in the circle, we can see that by moving along a specific direction corresponding to the slope of the plane, we can either reach a higher point or a lower point. However, since the circle has a finite boundary, that means there will be no other points further along that direction inside the circle \(R\). Thus, our highest and lowest points must necessarily be on the boundary of the circle.

Step 5: Conclusion

Therefore, by geometrical observation, we can conclude that the absolute maximum and minimum values of the function \(f(x, y) = ax + by + c\) over the closed bounded set \(R\) in \(\mathbb{R}^{2}\) must occur on the boundaries of \(R\).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions used in mathematics. It is represented in the form of \( f(x, y) = ax + by + c \), where \( a \), \( b \), and \( c \) are constants. In a linear function, the values change at a constant rate, which means that the graph of a linear function is always a straight line when plotted on a coordinate plane in \( \mathbb{R}^2 \).

• When plotted, linear functions create straight lines with constant slopes.
• In three dimensions \( \mathbb{R}^3 \), this concept extends to planes rather than lines.

The coefficients \( a \) and \( b \) dictate the slope of the line in 2D or the inclination of the plane in 3D. If both \( a \) and \( b \) are zero, the function becomes constant and doesn't represent a line or plane. However, in our context, at least one must be non-zero to maintain the linear relationship. Linear functions are fundamental in analyzing relationships where one variable changes with respect to others in a predictable way.

Bounded Sets

A set is termed as "bounded" if it has finite boundaries within which all of its points lie. In our problem, the set \( R \) is described as closed and bounded in \( \mathbb{R}^2 \). This signifies that \( R \) consists of all its limit points and is compact, meaning it is both contained and complete within its boundary.

• Closed sets contain their boundary points, ensuring no points exist just outside of the set boundary.
• Bounded sets do not extend to infinity in any direction, providing constraints to consider for analysis.

In practical terms, this means that one can think of \( R \) as a shape (like a circle, rectangle, or polygon) with finite size. The characteristics of being closed and bounded make \( R \) suitable for exploring maxima and minima because the finite boundary allows each extreme value to occur either precisely on the edge or vertex of the set.

Geometric Interpretation

Geometric interpretation involves visualizing mathematical concepts as shapes or lines, aiding in comprehending complex relationships. For linear functions such as \( f(x, y) = ax + by + c \), the geometric interpretation is crucial for determining where maximum and minimum values occur within a bounded set \( R \).
Imagine a linear function like a plane hovering over a shape on a map, which is the set \( R \). When visualized:

• The intersections of this plane with the boundary of \( R \) represent potential points where maximum or minimum values can be examined.
• Because the plane inclines in a specific direction determined by \( a \) and \( b \), the most extreme values in terms of height (or depth) manifest at the boundary points of set \( R \).

Essentially, since the plane does not swerve inwards or outwards from the direction determined by \( a \) and \( b \), the function's peaks or troughs are restricted to the limits of the physical boundary \( R \). This explains why for closed and bounded sets, as it extends infinitely in a direction, the highest or lowest points of the function must line up along these edges, offering a neat geometric solution to identifying the extrema.