Problem 22 - Thomas Calculus

Question

In Exercises \(21-36,\) find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$ f(x)=-x-4, \quad-4 \leq x \leq 1 $$

Step-by-Step Solution

Answer

The absolute maximum is 0 at \((-4, 0)\), and the absolute minimum is -5 at \((1, -5)\).

Step 1: Identify the Function and Interval

The given function is \( f(x) = -x - 4 \), and the interval is \(-4 \leq x \leq 1\). Our task is to find the absolute maximum and minimum values of this function within the given interval.

Step 2: Evaluate the Function at End Points

To find absolute extrema, we evaluate the function at the endpoints of the interval. Calculate \( f(x) \) at \( x = -4 \) and \( x = 1 \).- \( f(-4) = -(-4) - 4 = 4 - 4 = 0 \)- \( f(1) = -(1) - 4 = -1 - 4 = -5 \)

Step 3: Compare and Determine Extrema

Compare the values obtained from evaluating at the endpoints. The largest value is the absolute maximum, and the smallest is the absolute minimum.- Absolute maximum: \( f(-4) = 0 \)- Absolute minimum: \( f(1) = -5 \)

Step 4: Identify Coordinates of Extrema

Identify the coordinates on the graph where the extrema occur. - The absolute maximum point is \((-4, 0)\)- The absolute minimum point is \((1, -5)\)

Step 5: Graph the Function with Extrema Points

Draw the graph of the function \( f(x) = -x - 4 \) from \( x = -4 \) to \( x = 1 \). Mark the points \((-4, 0)\) and \((1, -5)\) on the graph as the locations of the absolute maximum and minimum, respectively.

Key Concepts

Function Evaluation

When it comes to determining the absolute extrema of a function, the evaluation at specific points is crucial. Function evaluation is the process of obtaining the function's output, or value, given a specific input, which is usually an x-value. In the original exercise, we evaluate the linear function \( f(x) = -x - 4 \) at the endpoints of the interval \([-4, 1]\). This gives us outputs at \( x = -4 \) and \( x = 1 \).

• At \( x = -4 \), the function evaluates as \( f(-4) = 0 \).
• At \( x = 1 \), the function evaluates as \( f(1) = -5 \).

These evaluations help us identify the extrema by comparing the results and finding which is maximum or minimum.

Graphing Functions

Graphing functions provides a visual way to understand how a function behaves over a specific interval. For the function \( f(x) = -x - 4 \), we need to create a graph between \( x = -4 \) and \( x = 1 \). A linear function like this results in a straight line graph. To graph, we plot points using the function evaluation results from the endpoints:

• The point \((-4, 0)\) is plotted for \( x = -4 \).
• The point \((1, -5)\) is plotted for \( x = 1 \).

Connect these two points with a straight line. The slope of the line will be \(-1\), indicating a downward slope as you move from left to right. This visual cue expresses how the function decreases in value over the interval.

Endpoints

Endpoints are crucial in analyzing functions over a closed interval. They mark the beginning and end of the interval over which we analyze the function. In this case, the given endpoints are \( x = -4 \) and \( x = 1 \).Evaluating a function at its endpoints allows us to determine potential extrema:

• Endpoints represent the boundary of our interval.
• Extrema, such as maximum and minimum values, are often located at these points.

Thus, evaluating the function at these specific values is a necessary step to fully understand the behavior of the function across the interval.

Coordinates of Extrema

Finding the coordinates of extrema is the process of identifying the specific points where a function reaches its highest or lowest value within an interval. For the function \( f(x) = -x - 4 \), we evaluated the function at both endpoints and found:

• The absolute maximum value occurs at the coordinate \((-4, 0)\).
• The absolute minimum value exists at the coordinate \((1, -5)\).

These coordinates provide exact locations on the graph where the function's extremum values occur, enabling us to graphically represent these critical points and understand the overall trend of the function.