Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the global maximum value and global minimum value of \(f\) on \(S\) and indicate where each occurs. \(\begin{aligned} \text f(x, y) &=3 x+4 y ; \\ S=\\{(x, y): 0&\leq x \leq 1,-1 \leq y \leq 1\\} \end{aligned}\)

Step-by-Step Solution

Answer

Maximum: 7 at (1, 1); Minimum: -4 at (0, -1).

Step 1: Identify the Function and Constraint Set

We are given the function \( f(x,y) = 3x + 4y \) and the constraint set \( S = \{(x, y): 0 \leq x \leq 1, -1 \leq y \leq 1\} \). Our task is to find the global maximum and minimum values of \( f \) within \( S \).

Step 2: Analyze Vertices of the Region

Since \( S \) is a rectangular region, the extreme values of \( f \) must occur at the vertices of this rectangle. The vertices of the region \( S \) are \((0, -1), (0, 1), (1, -1), (1, 1)\).

Step 3: Evaluate the Function at Each Vertex

Calculate \( f(x, y) \) at each vertex: - At \((0, -1)\), \( f(0, -1) = 3(0) + 4(-1) = -4 \).- At \((0, 1)\), \( f(0, 1) = 3(0) + 4(1) = 4 \).- At \((1, -1)\), \( f(1, -1) = 3(1) + 4(-1) = -1 \).- At \((1, 1)\), \( f(1, 1) = 3(1) + 4(1) = 7 \).

Step 4: Determine Global Maximum and Minimum

The maximum value among the vertices is \( f(1, 1) = 7 \) at \((1, 1)\), and the minimum value is \( f(0, -1) = -4 \) at \((0, -1)\).

Key Concepts

Global Maximum

In optimization, finding the global maximum involves determining the highest possible value of a function across a defined domain. In this context, the global maximum of the function \( f(x, y) = 3x + 4y \) within the constraint set \( S \) is evaluated. During this process, the function is analyzed at key points within the rectangular region \( S \).

These points are often the vertices or boundaries of the region. Specifically, for the rectangle defined by \( S = \{(x, y): 0 \leq x \leq 1, -1 \leq y \leq 1\} \), the vertices are \((0, -1), (0, 1), (1, -1), (1, 1)\).

When the function is evaluated at these points, the global maximum is identified as 7, which occurs at the vertex (1, 1). Identifying this point as the location of the global maximum involves checking all the potential critical points and ensuring no other points yield higher values within the region.

Global Minimum

The global minimum is the lowest value that a function attains over a specified domain. Similar to finding a global maximum, determining this requires evaluating the function at key locations within the defined region.

The rectangular region \( S = \{(x, y): 0 \leq x \leq 1, -1 \leq y \leq 1\} \) narrows down the possible range for both \( x \) and \( y \). Thus, evaluating the function \( f(x, y) = 3x + 4y \) at vertices is essential for finding the absolute minimum value.

The calculations reveal that the global minimum is -4, observed at the vertex (0, -1). Confirming this minimum involves checking that every other evaluated point does not result in a value lower than -4 within the rectangle.

Rectangular Region

The concept of a rectangular region is crucial when it comes to defining the domain within which the maximum and minimum values are to be determined. A rectangular region in this context refers to a set of possible \( (x, y) \) coordinate points bounded within a rectangle on the Cartesian plane.

Specifically, the set \( S = \{(x, y): 0 \leq x \leq 1, -1 \leq y \leq 1\} \) implies that \( x \) ranges from 0 to 1, while \( y \) ranges from -1 to 1. This rectangle includes the vertices or corner points \((0, -1), (0, 1), (1, -1), (1, 1)\), which are key locations for evaluating the function.

When solving optimization problems, recognizing that extreme values often occur at these boundaries helps simplify the process and efficiently find global maxima or minima.

Function Evaluation

Function evaluation is the process of calculating the value of a given function at specific points. In optimization problems, especially when dealing with multidimensional inputs, this step is critical in identifying extrema like global maxima and minima.

For the function \( f(x, y) = 3x + 4y \), evaluating at the vertices of the region \( S \) involves substituting the \( x \) and \( y \) coordinates of each vertex into the equation.

The evaluations at the vertices are as follows:

• At \((0, -1)\), \( f(0, -1) = -4 \)
• At \((0, 1)\), \( f(0, 1) = 4 \)
• At \((1, -1)\), \( f(1, -1) = -1 \)
• At \((1, 1)\), \( f(1, 1) = 7 \)

By performing these calculations, we ascertain the global extrema within the specified region. It is a systematic approach that ensures all critical points, particularly at the boundaries, are considered for analysis.