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12th Edition

Ronald J. Harshbarger + 1 other

Publisher: Cengage Learning

ISBN: 9781337625340

Chapter 10.1, Problem 36E

Textbook Problem

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In Problems 31-36, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function.

To determine

**To calculate:** The critical values, critical points, the relative maxima, relative minima, or horizontal points of inflection of the function

**Given Information:**

The provided function is

**Formula Used:**

To obtain the relative maxima and minima of a function,

1. Find the first derivative of the function.

2. Set the derivative equal to

3. Evaluate

(a) If

(b) If

**Calculation:**

Consider the provided function

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph.

Consider the provided derivative,

Put the value of

Evaluate the values of *x* from the equation:

Hence, the value(s) of *x* are

Evaluate the values of the original functions with the critical values:

Put

Hence,

Put

Hence,

Test the derivative at the left and right sides of all the critical points

Mathematical Applications for the Management, Life, and Social Sciences

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(a)...Ch. 10.1 - For each function and graph in Problems 11-14
(a)...Ch. 10.1 - For each function and graph in Problems 11-14
(a)...Ch. 10.1 - For each function and graph in Problems 11-14
(a)...Ch. 10.1 - For each function in Problems 15-20
(a) find
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(a) find
(b)...Ch. 10.1 - For each function in Problems 15-20
(a) find
(b)...Ch. 10.1 - For each function in Problems 15-20
(a) find
(b)...Ch. 10.1 - For each function in Problems 15-20
(a) find
(b)...Ch. 10.1 - For each function in Problems 15-20
(a) find
(b)...Ch. 10.1 - For each function and graph in Problems 21-24
(a)...Ch. 10.1 - For each function and graph in Problems 21-24
(a)...Ch. 10.1 - For each function and graph in Problems 21-24
(a)...Ch. 10.1 - For each function and graph in Problems 21-24
(a)...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - For each function in Problems 25-30, find the...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 31-36, both a function and its...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In Problems 37-42, use the derivative to locate...Ch. 10.1 - In each of Problems 43-46, a graph of is given....Ch. 10.1 - In each of Problems 43-46, a graph of is given....Ch. 10.1 - In each of Problems 43-46, a graph of is given....Ch. 10.1 - In each of Problems 43-46, a graph of is given....Ch. 10.1 - In Problems 47 and 48, two graphs are given. One...Ch. 10.1 - In Problems 47 and 48, two graphs are given. One...Ch. 10.1 - 49. Advertising and sales Suppose that the daily...Ch. 10.1 - 50. Pricing and sales Suppose that a chain of auto...Ch. 10.1 - 51. Productivity A time study showed that, on...Ch. 10.1 - 52. Production Analysis of daily output of a...Ch. 10.1 - 53. Production costs Suppose that the average...Ch. 10.1 - 54. Average costs Suppose the average costs of a...Ch. 10.1 - 55. Marginal revenue Suppose the weekly marginal...Ch. 10.1 - 56. Earnings Suppose that the rate of change of...Ch. 10.1 - 57. Revenue The weekly revenue of a certain...Ch. 10.1 - 58. Medication Suppose that the concentration C of...Ch. 10.1 - 59. Candidate recognition Suppose that the...Ch. 10.1 - 60. Medication The number of milligrams x of a...Ch. 10.1 - 61. Worldwide cell phone subscriberships In 2013,...Ch. 10.1 - 62. Economic dependency ratio The economic...Ch. 10.1 - 63. Modeling China’s labor pool The following...Ch. 10.1 - 64. Modeling Energy from crude oil The table shows...Ch. 10.1 - 65. Modeling Employment in manufacturing The table...Ch. 10.2 - 1. If then is concave__________.
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In Problems 11-14, a function and its graph are...Ch. 10.2 - In Problems 11-14, a function and its graph are...Ch. 10.2 - In Problems 11-14, a function and its graph are...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - Find the relative maxima, relative minima, and...Ch. 10.2 - In Problems 21-24, a function and its first and...Ch. 10.2 - In Problems 21-24, a function and its first and...Ch. 10.2 - In Problems 21-24, a function and its first and...Ch. 10.2 - In Problems 21-24, a function and its first and...Ch. 10.2 - In Problems 25 and 26, a function and its graph...Ch. 10.2 - In Problems 25 and 26, a function and its graph...Ch. 10.2 - In Problems 27 and 28, and its graph are given....Ch. 10.2 - In Problems 27 and 28, and its graph are given....Ch. 10.2 - In Problems 29 and 30, use the graph shown in the...Ch. 10.2 - In Problems 29 and 30, use the graph shown in the...Ch. 10.2 - In Problems 31 and 32, a graph is given. Tell...Ch. 10.2 - In Problems 31 and 32, a graph is given. Tell...Ch. 10.2 - 33. Productivity—diminishing returns The figure is...Ch. 10.2 - 34. Population growth The figure shows the growth...Ch. 10.2 - 35. Advertising and sales The figure shows the...Ch. 10.2 - 36. Oxygen purity The figure shows the oxygen...Ch. 10.2 - 37. Production Suppose that the total number of...Ch. 10.2 - 38. Poiscuille's law—velocity of blood According...Ch. 10.2 - 39. Advertising and sales—diminishing returns...Ch. 10.2 - 40. Oxygen purity—diminishing returns Suppose that...Ch. 10.2 - 41. Energy use Energy use per capita indexed to...Ch. 10.3 - 1. True or false: If is the total revenue...Ch. 10.3 - 2. If , form , the average cost function, and find...Ch. 10.3 - 3. (a) If gives the demand function in a monopoly...Ch. 10.3 - 4. If are the total revenue and total cost (both...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - In Problems 1-6, find the absolute maxima and...Ch. 10.3 - 7. (a) If the total revenue function for a hammer...Ch. 10.3 - 8. (a) If the total revenue function for a blender...Ch. 10.3 - 9. If the total revenue function for a computer is...Ch. 10.3 - 10. A firm has total revenues given by
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(a) A square piece of cardboard 36...Ch. 10.4 - 29. Revenue The owner of an orange grove must...Ch. 10.4 - 30. Minimum material
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(a) find all critical values,...Ch. 10 - In Problems 5-10
(a) find all critical values,...Ch. 10 - In Problems 5-10
(a) find all critical values,...Ch. 10 - In Problems 5-10
(a) find all critical values,...Ch. 10 - In Problems 5-10
(a) find all critical values,...Ch. 10 - In Problems 5-10
(a) find all critical values,...Ch. 10 - 11. Is the graph of concave up or concave down at...Ch. 10 - 12. Find intervals on which the graph of is...Ch. 10 - 13. Find the relative maxima, relative minima, and...Ch. 10 - In Problems 14 and 15, find any relative maxima,...Ch. 10 - In Problems 14 and 15, find any relative maxima,...Ch. 10 - 16. Given find the absolute maximum and minimum...Ch. 10 - 17. Given find the absolute maximum and minimum...Ch. 10 - In Problems 18 and 19, use the graphs to find the...Ch. 10 - In Problems 18 and 19, use the graphs to find the...Ch. 10 - In Problems 20 and 21, find any horizontal...Ch. 10 - In Problems 20 and 21, find any horizontal...Ch. 10 - In Problems 22-24
(a) find any horizontal and...Ch. 10 - In Problems 22-24
(a) find any horizontal and...Ch. 10 - In Problems 22-24
(a) find any horizontal and...Ch. 10 - In Problems 25 and 26, a function and its graph...Ch. 10 - In Problems 25 and 26, a function and its graph...Ch. 10 - In Problems 27 and 28, and its graph are...Ch. 10 - In Problems 27 and 28, and its graph are...Ch. 10 - In Problems 29 and 30, and its graph are...Ch. 10 - In Problems 29 and 30, and its graph are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - In Problems 31-36, cost, revenue, and profit are...Ch. 10 - 37. Marginal profit The figure shows the graph of...Ch. 10 - 38. Productivity—diminishing returns Suppose the...Ch. 10 - 39. Output The following figure shows a typical...Ch. 10 - 40. Revenue MMR II Extreme Bike Shop sells 54 of...Ch. 10 - 41. Profit If in Problem 40 the mountain bikes...Ch. 10 - 42. Profit Suppose that for a product in a...Ch. 10 - 43. Profit The monthly demand function for x units...Ch. 10 - 44. Profit Suppose that in a monopolistic market,...Ch. 10 - 45. Reaction to a drug The reaction R to an...Ch. 10 - 46. Productivity The number of parts produced per...Ch. 10 - 47. Population Population estimates show that the...Ch. 10 - 48. Night brightness Suppose that an observatory...Ch. 10 - 49. Product design A playpen manufacturer wants to...Ch. 10 - 50. Printing design A page is to contain 56 square...Ch. 10 - 51. Drug sensitivity The reaction R to an...Ch. 10 - 52. Per capita health care costs For the years...Ch. 10 - 53. Inventory cost model A company needs to...Ch. 10 - 54. Average cost Suppose the total cost of...Ch. 10 - 55. Market share Suppose a company’s percent share...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - Find the relative maxima, relative minima, points...Ch. 10 - In Problems 4-6, use the function .
4. Over what...Ch. 10 - In Problems 4-6, use the function .
5. Find the...Ch. 10 - In Problems 4-6, use the function .
6. Find the...Ch. 10 - 7. Find the absolute maximum and minimum for
on...Ch. 10 - 8. Find all horizontal and vertical asymptotes of...Ch. 10 - 9. Use the following graph of y = f(x) and the...Ch. 10 - 10. Use the following figure to complete parts...Ch. 10 - 11. If , and , what can we conclude about the...Ch. 10 - 12. The aged dependency ratio is defined as the...Ch. 10 - 13. The revenue function for a product is dollars...Ch. 10 - 14. The cost of producing x units of a product is...Ch. 10 - 15. A firm sells 100 TVs per month at $300 each,...Ch. 10 - 16. An open-top box is made by cutting squares...Ch. 10 - 17. A company estimates that it will need 784,000...

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