Problem 13 - Calculus for Biology and Medicine

Question

In Problems \(13-16\), the functions are defined on the rectangular domain $$ D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} $$ Find the absolute maxima and minima of \(f\) on \(D\). \(f(x, y)=2 x-y\)

Step-by-Step Solution

Answer

The absolute maximum is 3 at (1, -1) and the absolute minimum is -3 at (-1, 1).

Step 1: Understand the Domain

The domain given is a rectangle defined by \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\). This means we need to evaluate the function \(f(x, y) = 2x - y\) at these boundary points and within this area.

Step 2: Evaluate Function at Corners of Domain

The corners of the domain are points where the function may achieve extreme values. Evaluate \(f\) at these corners: \((-1, -1), (-1, 1), (1, -1), (1, 1)\). - \(f(-1, -1) = 2(-1) - (-1) = -2 + 1 = -1\)- \(f(-1, 1) = 2(-1) - 1 = -2 - 1 = -3\)- \(f(1, -1) = 2(1) - (-1) = 2 + 1 = 3\)- \(f(1, 1) = 2(1) - 1 = 2 - 1 = 1\)

Step 3: Consider Function on the Boundaries

The boundaries of the rectangle are where \(x = -1\), \(x = 1\), \(y = -1\), and \(y = 1\). Evaluate the function on these edges by varying the other variable:- On line \(x = -1\): Evaluate \(f(-1, y) = 2(-1) - y = -2 - y\). - Maxima and minima occur at endpoints where \(y = -1\) or \(y = 1\), which we evaluated in Step 2.- On line \(x = 1\): Evaluate \(f(1, y) = 2(1) - y = 2 - y\). - Maxima and minima occur at endpoints where \(y = -1\) or \(y = 1\), which we evaluated in Step 2.- On line \(y = -1\): Evaluate \(f(x, -1) = 2x + 1\). - Maxima and minima occur at endpoints where \(x = -1\) or \(x = 1\), which we evaluated in Step 2.- On line \(y = 1\): Evaluate \(f(x, 1) = 2x - 1\). - Maxima and minima occur at endpoints where \(x = -1\) or \(x = 1\), which we evaluated in Step 2.

Step 4: Identify the Absolute Maximum and Minimum

Based on the evaluations from Steps 2 and 3, the results are \(f(-1, -1) = -1\), \(f(-1, 1) = -3\), \(f(1, -1) = 3\), and \(f(1, 1) = 1\). Of these, the absolute maximum value is 3 at the point (1, -1) and the absolute minimum value is -3 at the point (-1, 1).

Key Concepts

Rectangular Domain

A rectangular domain is a part of the coordinate plane defined by the range of values that two variables, usually denoted as \(x\) and \(y\), can take. In this exercise, the domain is specified as \(-1 \leq x \leq 1\) and \(-1 \leq y \leq 1\).
This creates a rectangle that includes all points \((x, y)\) where both \(x\) and \(y\) satisfy these inequalities.

• The edges of the rectangle represent boundaries where extreme values of a function can occur.
• The corners of the rectangle are particularly important as they are locations where the simplest types of extreme values often present themselves.

Evaluating a given function within and on the boundary of this domain is essential to effectively find absolute extrema. Keep in mind, this domain allows us to limit where the search for the highest and lowest values of the function should focus.

Boundary Evaluation

Boundary evaluation involves checking the behavior of a function along the borders of its defined domain, which may involve lines or curves. For the exercise involving the rectangular domain \(D\), the boundary evaluation is particularly crucial.

• Boundaries are places where extreme values, like maximum or minimum, are likely to occur because changes outside of the boundary are not part of the domain.
• In a simple rectangle, these boundaries are the edges where \(x\) is constant \((-1, 1)\) and \(y\) is constant \((-1, 1)\).
• For each boundary line, evaluate the function by substituting the fixed variable into the function and then finding the critical points along the remaining one-variable that lies on the boundary.

Through such evaluation, we can determine the largest and smallest function values along each edge, leading to a complete understanding of potential extrema locations.

Two-variable Functions

Two-variable functions are expressions that involve two independent variables, such as \(x\) and \(y\), and are of the form \(f(x, y)\). The function given here, \(f(x, y) = 2x - y\), is linear, meaning it forms a plane in three-dimensional space.

• The function's behavior depends on changes in both variables, allowing us to analyze how peaks and valleys change across a surface.
• Understanding the contour plot of the function can provide insights into where expected maxima or minima occur.
• For linear functions, critical points are often more straightforward to identify, essentially appearing at the edges or corners of the domain.

Evaluating such functions involves not only looking at specific values but understanding how changes in \(x\) and \(y\) affect \(f(x, y)\) globally.