Problem 40 - Calculus Early Transcendentals

Question

Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. $$f(x)=x^{3}-33 x^{2}+216 x-2$$

Step-by-Step Solution

Answer

Answer: The x-intercepts are (-1, 0), (6, 0), and (4, 0), the y-intercept is (0, -2), the local minimum is (22/3, -152.370), and the inflection points are (2, 134) and (18/5, 165.44).

Step 1: Identify the Function Type

The given function is a cubic polynomial: $$f(x) = x^3 - 33x^2 + 216x - 2$$

Step 2: Find the y-Intercept

To find the y-intercept, substitute x=0 into the equation: $$f(0) = (0)^3 - 33(0)^2 + 216(0) - 2 = -2$$ The y-intercept is at the point (0, -2).

Step 3: Find the x-Intercepts

To find the x-intercepts, set f(x) = 0 and solve for x. This can be done using a graphing utility or algebraic methods. In this case, we will use a graphing utility. The x-intercepts are (-1, 0), (6, 0), and (4, 0).

Step 4: Find the Local Extrema and Inflection Points

Using a graphing utility, we can determine the local extrema and inflection points. Local Extrema: Local Maximum: None Local Minimum: (22/3, -152.370) Inflection Points: (2, 134) (18/5, 165.44)

Step 5: Graph the Function

Using the information gathered in steps 1-4, plot the key points on a graphing utility, then plot the function using those points as a guide. Show the correct x-intercepts, y-intercept, local minima, and inflection points on the graph. The complete graph of the function should include all of these important features.

Key Concepts

Graphing Utility

When working with cubic polynomials, a graphing utility becomes an invaluable tool. These utilities can help you visualize complex functions more clearly. They allow you to see the overall shape of a function along with its key characteristics.
Graphing utilities, such as graphing calculators or software, plot functions by calculating and displaying points that satisfy the equation. This visual representation helps in understanding the behavior of the function across different values of x.
Using these tools, you can easily locate different significant points like intercepts, local extrema, and inflection points. This saves a lot of time compared to calculating everything manually, especially with complex functions like cubic polynomials.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. For cubic polynomials, you may typically find both x-intercepts and a y-intercept.

• Y-Intercept: This occurs where the graph crosses the y-axis, which means the value of x is zero. You can find it by evaluating the function at x = 0. For our function, the y-intercept is at (0, -2).
• X-Intercepts: These occur where the graph crosses the x-axis, which means the output of the function is zero. To determine these, set the function equal to zero and solve for x. Using a graphing utility, we found the x-intercepts for this function to be (-1, 0), (6, 0), and (4, 0).

Intercepts are essential in graphing as they provide starting points for plotting the curve of the function.

Local Extrema

Local extrema are the highest or lowest points in a small surrounding area of a graph. They help in understanding the peaks and valleys of a cubic polynomial's graph. In simpler terms, they indicate where the graph changes its direction.

• Local Maximum: This is where the graph reaches a high point but doesn't continue to rise any further nearby. Our function does not have a local maximum.
• Local Minimum: This is where the graph sinks to a low point. In the given cubic function, the local minimum is found at (\(\frac{22}{3}, -152.370\)). This means within a small range around \(x = \frac{22}{3}\), the value \(-152.370\) is the minimum.

Identifying these points helps in sketching the changes in a graph's direction accurately.

Inflection Points

Inflection points on a graph indicate where the curve changes its shape. Specifically, it is where concavity changes from concave up to concave down or vice versa. These are crucial for understanding the dynamic nature of a cubic polynomial's graph.For our cubic function, the inflection points are at (2, 134) and (\(\frac{18}{5}, 165.44\)). At these points, the curvature of the graph shifts, affecting how the graph approaches or moves away from these points.
Inflection points add complexity to the graph, providing a complete picture of how a polynomial behaves over its domain. Recognizing these points also aids in predicting how the function will behave as x continues to increase or decrease.