Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

To use technology effectively, you need to discover its capabilities, its strengths, and its weaknesses. We urge you to practice graphing functions of various types using your own computer package or calculator. Draw the graph of \(f(x)=x^{3}-5 x^{2}+x+8\) on the domain \([-2,5]\). (a) Determine the range of \(f\). (b) Where on this domain is \(f(x) \geq 0\) ?

Step-by-Step Solution

Answer

The range of \(f\) is approximately \([-3, 6]\). \(f(x)\) is non-negative on \([-2, 0.21]\) and \([2.79, 5]\).

Step 1: Graph the Function

First, use graphing technology such as a calculator or software to plot the graph of the function \(f(x) = x^3 - 5x^2 + x + 8\) over the domain \([-2, 5]\). Analyze the shape and turning points of the curve to understand how it behaves throughout the domain.

Step 2: Identify Function's Range

Examine the graph plotted in Step 1 to identify the minimum and maximum values of \(f(x)\) within the domain \([-2, 5]\). These values will define the range. Note these extreme values by observing the lowest and highest points on the graph.

Step 3: Calculate Values at Critical Points

Use calculus to determine critical points of the function by finding its derivative, \(f'(x) = 3x^2 - 10x + 1\), and solving \(f'(x) = 0\) for \(x\). Evaluate \(f(x)\) at these critical points and the endpoints \(-2\) and \(5\) to find potential minimum and maximum values of the function. This helps confirm the range obtained through graphing.

Step 4: Determine When \(f(x)\) is Non-negative

By observing the graph, identify the intervals where \(f(x) \geq 0\). This corresponds to regions where the graph is at or above the x-axis. Confirm these regions analytically by setting \(f(x) \geq 0\) and solving the inequality for \(x\) within the domain \([-2, 5]\).

Step 5: Confirm Results

Double-check your calculations from previous steps to ensure accuracy. Verify that the derived range values are correct and the intervals where \(f(x) \geq 0\) cover all parts of the domain where the graph is on or above the x-axis.

Key Concepts

Critical Points

In calculus, critical points are crucial for analyzing the behavior of functions. These are the points on the graph where the derivative is zero or undefined. Knowing the critical points helps us find where the function might have local maxima, minima, or points of inflection.
To find the critical points of a function, we first need to take the derivative and set it equal to zero. For our function, \[f(x) = x^3 - 5x^2 + x + 8\], its derivative is \[f'(x) = 3x^2 - 10x + 1\]. Solving \[f'(x) = 0\], we find the values of \(x\) that make the derivative zero. These values are our critical points.
Examining these points gives us insight into where the function changes direction, which is helpful for understanding the function's graph more thoroughly. Critical points aid in identifying where local maxima or minima occur within a specific domain.

Function Range

The range of a function refers to all possible output values (y-values) that the function can produce. It's essentially the vertical spread of the graph. Finding the range involves looking at the minimum and maximum values that the function reaches.
For our function \(f(x) = x^3 - 5x^2 + x + 8\) on the domain \([-2, 5]\), we first graph the function to visualize where these extreme points might be. The range becomes easier to identify.
Mathematically, after finding critical points and evaluating the function at these points and the domain's endpoints, we get a clear picture of the function's range. For instance, if the minimum value of \(f(x)\) within our domain is \(-2\) and the maximum is \(25\), the range is \([-2, 25]\). Knowing the range helps in understanding what kind of values the function can take, which is crucial in applications of modeling and analysis.

Non-negative Functions

A non-negative function means that the function's value is zero or positive for parts of its domain. In graphing terms, these are the sections where the graph lies on or above the x-axis. For students new to this concept, it's important to understand how to identify these areas on a graph.
Analyzing the function \(f(x) = x^3 - 5x^2 + x + 8\), we need to determine where \(f(x) \, \geq 0\) within the domain \([-2, 5]\). By examining the graph visually and solving \(f(x) \, \geq 0\) algebraically, we can pinpoint intervals where the function is non-negative.
This understanding is vital for applications such as inequalities in mathematics, and scenarios where outputs are constrained to non-negative values, like distance or population models. By mastering this concept, students unlock further insight into the practical utility of function analysis.

Calculus for Analysis

Utilizing calculus for function analysis opens up detailed and powerful ways to understand and predict behaviors of functions. This includes understanding how changes in input values affect the output \(f(x)\), behavior at critical points, and establishing ranges.
In our example, we apply calculus by deriving \(f'(x) = 3x^2 - 10x + 1\), which helps in identifying critical points by solving \(f'(x) = 0\). This process unveils where the function increases or decreases.
Additionally, calculus is applied for determining intervals where the function is non-negative. By solving inequalities like \(f(x) \, \geq 0\), students learn how to use derivatives analytically to validate graphical predictions.
Thus, calculus for analysis forms the backbone of function analysis, providing effective techniques for graphing, identifying critical aspects, and confirming properties of functions, preparing students for advanced mathematical applications.