Explanation Given Information: The provided function is, f ( x ) = x 3 + 6 x 2 + 9 x + 3 Formula used: A vertical asymptote of a function f ( x ) is a line x = a such that f ( a ) = ∞ . A vertical asymptote of a rational function h ( x ) = f ( x ) g ( x ) is x = a where g ( a ) = 0 and f ( a ) ≠ 0 . A horizontal asymptote of a function f ( x ) is a line y = b such that lim x → ∞ f ( x ) = b or lim x → − ∞ f ( x ) = b . A horizontal asymptote of a rational function h ( x ) = f ( x ) g ( x ) is 1. A line y = 0 if the degree of the numerator is less than the degree of the denominator. 2. The line y is ratio of the leading coefficients if the degree of the numerator is equal to the degree of the denominator. 3. Does not exist if the degree of the numerator is greater than the degree of the denominator. To find relative maxima and minima of a function, 1. Set the first derivative of the function to zero, f ' ( x ) = 0 , to find the critical values of the function. 2. Substitute the critical values into f ( x ) and calculate the critical points. 3. Calculate the sign of the function to the left and right of the critical values. (a) If f ′ ( x ) < 0 and f ′ ( x ) > 0 on the left and right side respectively of a critical value, then the point is relative minimum. (b) If f ′ ( x ) > 0 and f ′ ( x ) < 0 on the left and right side respectively of a critical value, then the point is relative maximum. Calculation: Consider the provided function, f ( x ) = x 3 + 6 x 2 + 9 x + 3 Now, calculate the first derivative of above with respect to x and solve: f ′ ( x ) = 3 x 2 + 12 x + 9 = 3 ( x 2 + 4 x + 3 ) = 3 ( x 2 + 3 x + x + 3 ) = 3 ( x + 3 ) ( x + 1 ) Now, to obtain the critical values, set f ′ ( x ) = 0 as, 3 ( x + 3 ) ( x + 1 ) = 0 Thus, either ( x + 1 ) = 0 or ( x + 3 ) = 0 . First consider ( x + 1 ) = 0 . Subtract 1 from both sides. x + 1 − 1 = 0 − 1 x = − 1 And, Subtract 3 from both sides. x + 3 − 3 = 0 − 3 x = − 3 Thus, the critical values of the function are at x = − 1 and x = − 3 . Now, substitute − 1 for x in the equation f ( x ) = x 3 + 6 x 2 + 9 x + 3 , f ( x ) = ( − 1 ) 3 + 6 ( − 1 ) 2 + 9 ( − 1 ) + 3 = − 1 + 6 − 9 + 3 = − 1 Substitute − 3 for x in the equation f ( x ) = x 3 + 6 x 2 + 9 x + 3 , f ( x ) = ( − 3 ) 3 + 6 ( − 3 ) 2 + 9 ( − 3 ) + 3 = − 27 + 54 − 27 + 3 = 3 Thus, the critical points are ( − 1 , − 1 ) and ( − 3 , 3 ) . Now, calculate the sign of the y ′ on either side of the critical point. First consider x = − 1 . Calculate f ′ ( x ) at x = − 2 . f ′ ( x ) = 3 ( x + 3 ) ( x + 1 ) = 3 ( − 2 + 3 ) ( − 2 + 1 ) = 3 ( 1 ) ( − 1 ) = − 3 Thus, f ′ ( x ) < 0 on left of x = − 1

WebAssign Printed Access Card for Harshbarger/Reynolds' Mathematical Applications for the Management, Life, and Social Sciences, 12th Edition, Multi-Term

12th Edition

ISBN: 9781337630542

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 10, Problem 1T

To determine

To calculate: The relative minimum, relative maximum, points of inflection and asymptotes for the function f(x)=x3+6x2+9x+3 and sketch its graph.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10, Problem 55RE

Chapter 10, Problem 2T

Students have asked these similar questions

(c) Describe the distribution plan and show the total distribution cost. Optimal Solution Amount Cost $ 2000 Southern-Hamilton 200 Southern-Butler $ Southern-Clermont 300 4500 Northwest-Hamilton 200 $2400 Northwest-Butler 200 $3000 Northwest-Clermont $ Total Cost ક (d) Recent residential and industrial growth in Butler County has the potential for increasing demand by 100 units. (i) Create an updated distribution plan assuming Southern Gas becomes the preferred supplier. Distribution Plan with Southern Gas Amount Southern-Hamilton $ Cost × Southern-Butler x $ Southern-Clermont 300 $ 4500 Northwest-Hamilton 64 x Northwest-Butler $ × Northwest-Clermont 0 $0 Total Cost $ (ii) Create an updated distribution plan assuming Northwest Gas becomes the preferred supplier. Distribution Plan with Northwest Gas Southern-Hamilton Southern-Butler 0 Southern-Clermont Northwest-Hamilton Northwest-Butler Northwest-Clermont Total Cost Amount × x x +7 $0 Cost × $ × $ × +4 $ -/+ $ × ×

The distribution system for the Herman Company consists of three plants, two warehouses, and four customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows. Warehouse Plant Capacity 1 2 1 4 7 450 2 8 5 600 3 5 6 380 Customer demand and shipping costs per unit (in $) from each warehouse to each customer are as follows. Customer Warehouse 1 2 3 1 6 4 8 2 3 6 7 7 Demand 300 300 300 400 (a) Develop a network representation of this problem. (Submit a file with a maximum size of 1 MB.) Choose File No file chosen This answer has not been graded yet. (b) Formulate a linear programming model of the problem. (Let Plant 1 be node 1, Plant 2 be node 2, Plant 3 be node 3, Warehouse 1 be node 4, Warehouse 2 be node 5, Customer 1 be node 6, Customer 2 be node 7, Customer 3 be node 8, and Customer 4 be node 9. Express your answers in the form x;;, where x,; represents the number of units shipped from node i to node j.) Min 4x14+8x24+5x34+7x15 +5x25…

A linear programming computer package is needed. Hanson Inn is a 96-room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. A large profesional organization has scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least 50% of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two-night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two-night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are…

Chapter 10 Solutions

WebAssign Printed Access Card for Harshbarger/Reynolds' Mathematical Applications for the Management, Life, and Social Sciences, 12th Edition, Multi-Term

Ch. 10.1 - 1. The x- values of critical points are found...Ch. 10.1 - Prob. 2CP Ch. 10.1 - Prob. 3CP Ch. 10.1 - Prob. 4CP Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - 3. Use the graph of in Problem 1 to identify at...Ch. 10.1 - 4. Use the graph of in Problem 2 to identify at...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The relative minimum, relative maximum , points of inflection and asymptotes for the function f ( x ) = x 3 + 6 x 2 + 9 x + 3 and sketch its graph.

Solution Summary: The author calculates the relative minimum, relative maximum, points of inflection and asymptotes for the function f(x) and sketch its graph.

Concept explainers

Videos

To calculate: The relative minimum, relative maximum, points of inflection and asymptotes for the function f(x)=x3+6x2+9x+3 and sketch its graph.

Want to see the full answer?

Chapter 10 Solutions