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All Textbook Solutions for Mathematical Applications for the Management, Life, and Social Sciences
For each function in Problems 43-46, do the following.
(a) Find.
(b) Graph both and using a graphing calculator.
(c) Use the graph of to identify x-values where .
(d) Use the graph of to identify x-values where has a maximum or minimum point, where the graph of is rising, and where the graph of is falling.
46.
47E48. Revenue The total revenue, in dollars, for a commodity is described by the function
(a) What is the marginal revenue when 40 units are sold?
(b) Interpret your answer to part (a).
49E50. Capital investment and output The monthly output of a certain product is
where x is the capital investment in millions of dollars. Find dQ/dx, which can be used to estimate the effect on the output if an additional capital investment of $1 million is made.
51. Demand The demand for q units of a product depends on the price p (in dollars) according to
Find and explain the meaning of the instantaneous rate of change of demand with respect to price when the price is
(a) $25. (b) $100.
52E53. Cost and average cost Suppose that the total cost function, in dollars, for the production of x units of a product is given by
Then the average cost of producing x items is
(a) Find the instantaneous rate of change of average cost with respect to the number of units produced at any level of production.
(b) Find the level of production at which this rate of change equals zero.
(c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost. What do you notice?
54. Cost and average cost Suppose that the total cost function, in dollars, for a certain commodity is given by
where x is the number of units produced.
(a) Find the instantaneous rate of change of the average cost
for any level of production.
(b) Find the level of production where this rate of change equals zero.
(c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost. What do you notice?
55. Cost-benefit Suppose that tor a certain city, the cost C, in dollars, of obtaining drinking water that contains p percent impurities (by volume) is given by
(a) Find the rate of change of cost with respect to p when impurities account for 10% (by volume).
(b) Write a sentence that explains the meaning of your answer in part (a).
56E57. Wind chill One form of the formula that meteorologists use to calculate wind chill temperature WC is
where s is the wind speed in mph and t is the actual air temperature in degrees Fahrenheit. Suppose temperature is constant at .
(a) Express wind chill WC as a function of wind speed s.
(b) Find the rate of change of wind chill with respect to wind speed when the wind speed is 25 mph.
(c) Interpret your answer to part (b).
58E1. True or false: The derivative of the product of two functions is equal to the product of the derivatives of the two functions.
2. True or false: The derivative of the quotient of two functions is equal to the quotient of the derivatives of the two functions.
3CP4CP5. If , where c is a constant, does finding y' require the Quotient Rule? Explain.
2EIn Problems 1-4, find the derivative and simplify.
2.
4EIn Problems 1-4, find the derivative and simplify.
4.
In Problems 5-8, find the derivative but do not simplify your answer.
5.
In Problems 5-8, find the derivative but do not simplify your answer.
6.
In Problems 5-8, find the derivative but do not simplify your answer.
7.
In Problems 5-8, find the derivative but do not simplify your answer.
8.
In Problems 9 and 10, at each indicated point, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
9.
In Problems 9 and 10, at each indicated point, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
10.
In Problems 11-20, find the indicated derivatives and simplify.
11.
In Problems 11-20, find the indicated derivatives and simplify.
12.
In Problems 11-20, find the indicated derivatives and simplify.
13.
14E15E16EIn Problems 11-20, find the indicated derivatives and simplify.
17.
18E19E20EIn Problems 21 and 22, at the indicated point for each function, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
21. at (2, 1 )
In Problems 21 and 22, at the indicated point for each function, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
22.
In Problems 23-26, write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line.
23.
In Problems 23-26, write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line.
24. at
25E26E27E28E29E30E35E36E37E38. Use the Quotient Rule to show that the Powers of x Rule applies to negative integer powers. That is, show that , by finding the derivative of .
39E40. Cost-benefit If the cost C (in dollars) of removing p percent of the impurities from the wastewater in a manufacturing process is given by
find the rate of change of C with respect to p.
41. Revenue Suppose the revenue (in dollars) from the sale of x units of a product is given by
Find the marginal revenue when 49 units are sold. Interpret your result.
42. Revenue The revenue (in dollars) from the sale of x units of a product is given by
Find the marginal revenue when 149 units are sold. Interpret your result.
43E44E45. Response to a drug The reaction R to an injection of a drug is related to the dosage x (in milligrams) according to
where 1000 mg is the maximum dosage. If the rate of reaction with respect to the dosage defines the sensitivity to the drug, find the sensitivity.
46. Nerve response The number of action potentials produced by a nerve, t seconds after a stimulus, is given by
Find the rate at which the action potentials are produced by the nerve.
47. Test reliability If a test having reliability r is lengthened by a factor n, the reliability of the new test is given by
Find the rate at which R changes with respect to n.
48. Advertising and sales The sales of a product s (in thousands of dollars) are related to advertising expenses (in thousands of dollars) by
Find and interpret the meaning of the rate of change of sales with respect to advertising expenses when
(a) . (b) .
49. Candidate recognition Suppose that the proportion P of voters who recognize a candidate’s name t months after the start of the campaign is given by
(a) Find the rate of change of P when t = 6 and explain its meaning.
(b) Find the rate of change of P when t = 12 and explain its meaning.
(c) One month prior to the election, is it better for P'(t) to be positive or negative? Explain.
50E51. Wind chill Each winter the so-called “polar vortex” of deep frigid air plunges into the United States and sometimes brings record cold and dangerous wind chills. If s is the wind speed in miles per hour and , then the wind chill (in degrees Fahrenheit) for an air temperature of can be approximated by the function
(a) At what rate is the wind chill changing when the wind speed is 20 mph?
(b) Explain the meaning of your answer to part (a).
52. Response to injected adrenalin Experimental evidence has shown that the response y of a muscle is related to the concentration of injected adrenaline x according to the equation
where a and b are constants. Find the rate of change of response with respect to the concentration.
53E54. Emissions The table shows data for sulfur dioxide emissions from electricity generation (in millions of short tons) for selected years from 2000 and projected to 2035. These data can be modeled by the function
where x is the number of years past 2000.
(a) Find the function that models the rate of change of these emissions.
(b) Find and interpret
Year Short Tons (in millions)
2000 11.4
2005 10.2
2008 7.6
2015 4.7
2020 4.2
2025 3.8
2030 3.7
2035 3.8
Source: U.S. Department of Energy
55E56E
2. (a) If , find f'(x) by using the Power Rule (not the Quotient Rule).
(b) If , find f'(x) by using the Power Rule (not the Quotient Rule).
1E2E3E4EDifferentiate the functions in Problems 3-20.
3.
Differentiate the functions in Problems 3-20.
4.
Differentiate the functions in Problems 3-20.
5.
Differentiate the functions in Problems 3-20.
6.
Differentiate the functions in Problems 3-20.
7.
Differentiate the functions in Problems 3-20.
8.
Differentiate the functions in Problems 3-20.
9.
Differentiate the functions in Problems 3-20.
10.
Differentiate the functions in Problems 3-20.
11.
Differentiate the functions in Problems 3-20.
12.
Differentiate the functions in Problems 3-20.
13.
Differentiate the functions in Problems 3-20.
14.
Differentiate the functions in Problems 3-20.
15.
Differentiate the functions in Problems 3-20.
16.
Differentiate the functions in Problems 3-20.
17.
Differentiate the functions in Problems 3-20.
18.
Differentiate the functions in Problems 3-20.
19.
Differentiate the functions in Problems 3-20.
20.
At the indicated point, for each function in Problems 21-24, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
21.
At the indicated point, for each function in Problems 21-24, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
22.
At the indicated point, for each function in Problems 21-24, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
23.
At the indicated point, for each function in Problems 21-24, find
(a) the slope of the tangent line.
(b) the instantaneous rate of change of the function.
24.
In Problems 25-28, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.
In Problems 25-28, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.
In Problems 25-28, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.
In Problems 25-28, write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct.
28.
In Problems 29 and 30, complete the following for each function.
(a) Find f'(x).
(b) Check your result in part (a) by graphing both it and the numerical derivative of the function.
(c) Find x-values for which the slope of the tangent is 0.
(d) Find points (x, y) where the slope of the tangent is 0.
(e) Use a graphing utility to graph the function and locate the points found in part (d).
29.
In Problems 29 and 30, complete the following for each function.
(a) Find f'(x).
(b) Check your result in part (a) by graphing both it and the numerical derivative of the function.
(c) Find x-values for which the slope of the tangent is 0.
(d) Find points (x, y) where the slope of the tangent is 0.
(e) Use a graphing utility to graph the function and locate the points found in part (d).
30.
In Problems 31 and 32, do the following for each function .
(a) Find .
(b) Graph both and with a graphing utility.
(c) Determine x-values where .
(d) Determine x-values for which has a maximum or minimum point, where the graph is increasing, and where it is decreasing.
31.
In Problems 31 and 32, do the following for each function .
(a) Find .
(b) Graph both and with a graphing utility.
(c) Determine x-values where .
(d) Determine x-values for which has a maximum or minimum point, where the graph is increasing, and where it is decreasing.
32.
In Problems 33 and 34, find the derivative of each function.
33. (a) (b)
(c) (d)
In Problems 33 and 34, find the derivative of each function.
34. (a)
(b)
(c)
(d)
35. Ballistics Ballistics experts are able to identify the weapon that fired a certain bullet by studying the markings on the bullet. Tests are conducted by firing into a bale of paper. If the distance s, in inches, that the bullet travels into the paper is given by
for second, find the velocity of the bullet one-tenth of a second after it hits the paper.
36. Population of microorganisms Suppose that the population of a certain microorganism at time t (in minutes) is given by
Find the rate of change of population.
37. Revenue The revenue from the sale of a product is where x is the number of units sold. Find the marginal revenue when 100 units are sold. Interpret your result.
40E39. Pricing and sales Suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to
(a) What is the rate of change in sales volume when the price is $21?
(b) Interpret your answer to part (a).
40. Pricing and sales A chain of auto service stations has found that its monthly sales volume S (in thousands of dollars) is related to the price p (in dollars) of an oil change according to
(a) What is the rate of change of sales volume when the price is $44?
(b) Interpret your answer to part (a).
41. Demand Suppose that the demand tor q units of a product priced at $p per unit is described by
(a) What is the rate of change of price with respect to the quantity demanded when q = 49?
(b) Interpret your answer to part (a).
44EStimulus-response The relation between the magnitude of a sensation y and the magnitude of the stimulus x is given by
where k is a constant, is the threshold of effective stimulus, and n depends on the type of stimulus. Find the rate of change of sensation with respect to the amount of stimulus for each of Problems 42-44.
43. For the stimulus of warmth
46E45. Demand If the demand for q units of a product priced at $p per unit is described by the equation
find the rate of change of p with respect to q.
46. Advertising and sales The daily sales S (in thousands of dollars) attributed to an advertising campaign are given by
where t is the number of weeks the campaign runs. What is the rate of change of sales at (a) t = 8? (b) t = 10?
(c) Should the campaign be continued after the 10th week? Explain.
49E48. Data entry speed The data entry speed (in entries per minute) of a data clerk trainee is
where x is the number of hours of training he has had. What is the rate at which his speed is changing, and what does this rate mean when he has had
(a) 15 hours of training?
(b) 40 hours of training?
49. Investments If an IRA is a variable-rate investment for 20 years at rate r percent per year, compounded monthly, then the future value S that accumulates from an initial investment of $1000 is
What is the rate of change of S with respect to r, and what does it tell us if the interest rate is
(a) ? (b) ?
52E53E52. Energy use Energy use per dollar of GDP indexed to 1980 means that energy use for any year is viewed as a percent of the use per dollar of GDP in 1980. The following data show the energy use per dollar of GDP, as a percent, for selected years from 1985 and projected to 2035.
Energy Use per Dollar of GDP
Year Percent Year Percent
1985 83 2015 51
1990 79 2020 45
1995 75 2025 41
2000 67 2030 37
2005 60 2035 34
2010 56
Source: U.S. Department of Energy
These data can be modeled with the function
where E(t) is the energy use per dollar of GDP (indexed to 1980) and t is the number of years past 1980.
(a) Find E'(t).
(b) Use this model to find and interpret the instantaneous rates of change of energy use per dollar of GDP in 2000 and 2025.
(c) Use the data in the table to find an average rate of change that approximates the 2025 instantaneous rate.
55E56E1. If a function has the form , where n is a constant, we begin to find the derivative by using the_____ Rule and then use the_____ Rule to find the derivative of .
2CP3CPFind the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
2.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
3.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
4.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
5.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
6.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
7.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
8.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
9.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
10.
Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
11.
12E13E14E15E16E17E18E19E20E21E22E23E24E25EFind the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only.
26.
27E28E29E30E31E32EIn Problems 33 and 34, find the derivative of each function.
33. (a)
(b)
(c)
(d)
34E35E36. Revenue Suppose that the revenue function for a certain product is given by
where x is in thousands of units and R is in thousands of dollars.
(a) Find the marginal revenue when 2000 units are sold.
(b) How is revenue changing when 2000 units are sold?
37. Revenue Suppose that the revenue in dollars from the sale of x campers is given by
(a) Find the marginal revenue when 10 units are sold.
(b) How is revenue changing when 10 units are sold?
38E39E40. Demand Suppose that the demand function for q units of an appliance priced at $p per unit is given by
Find the rate of change of price with respect to the number of appliances.
41E42. Advertising and sales Suppose that sales (in thousands of dollars) are directly related to an advertising campaign according to
where t is the number of weeks of the campaign.
(a) Find the rate of change of sales after 3 weeks.
(b) Interpret the result in part (a).
43. Advertising and sales An inferior product with an extensive advertising campaign does well when it is released, but sales decline as people discontinue use of the product. If the sales S (in thousands of dollars) after t weeks are given by
what is the rate of change of sales when t = 9? Interpret your result.
44. Advertising and sales An excellent film with a very small advertising budget must depend largely on word-of-mouth advertising. If attendance at the film after t weeks is given by
what is the rate of change in attendance and what does it mean when ?
45ESuppose that the distance a particle travels is given by
where s is in feet and x is in seconds.
1. Find the function that describes the velocity of this particle.
2CP3CP4CP1E2EIn Problems 1-6, find the second derivative.
4E5E6EIn Problems 7-12, find the third derivative.
In Problems 7-12, find the third derivative.
In Problems 7-12, find the third derivative.
In Problems 7-12, find the third derivative.
In Problems 7-12, find the third derivative.
In Problems 7-12, find the third derivative.
In Problems 13-24, find the indicated derivative.
In Problems 13-24, find the indicated derivative.
15E16EIn Problems 13-24, find the indicated derivative.
17. Find
18EIn Problems 13-24, find the indicated derivative.
19. Find
20E21E22E23E24E25E26E In Problems 27-30, use the numerical derivative feature of a graphing calculator to approximate the given second derivatives.
27.
28E29E30E31E32E33. Acceleration A particle travels as a function of time according to the formula
where s is in meters and t is in seconds. Find the acceleration of the particle when t = 2.
34. Acceleration If the formula describing the distance s (in feet) an object travels as a function of time (in seconds) is
what is the acceleration of the object when t = 4?
35. Revenue The revenue (in dollars) from the sale of x units of a certain product can be described by
Find the instantaneous rate of change of the marginal revenue.
36. Revenue Suppose that the revenue (in dollars) from the sale of a product is given by
where x is the number of units sold. How fast is the marginal revenue changing when x = 100?
37. Sensitivity When medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled by
where c is a positive constant and m is the amount of medicine absorbed into the blood (Source: R. M. Thrall et al., Some Mathematical Models in Biology, U.S. Department of Commerce, 1967). The sensitivity to the medication is defined to be the rate of change of reaction R with respect to the amount of medicine m absorbed in the blood.
(a) Find the sensitivity.
(b) Find the instantaneous rate of change of sensitivity with respect to the amount of medicine absorbed in the blood.
(c) Which order derivative of reaction gives the rate of change of sensitivity?
38. Photosynthesis The amount of photosynthesis that takes place in a certain plant depends on the intensity of light x according to the equation
(a) Find the rate of change of photosynthesis with respect to the intensity.
(b) What is the rate of change when ? When ?
(c) How fast is the rate found in part (a) changing when ? When ?
39. Revenue The revenue (in thousands of dollars) from the sale of a product is
where x is the number of units sold.
(a) At what rate is the marginal revenue changing when the number of units being sold is 25?
(b) Interpret your result in part (a).
40E41. Advertising and sales The daily sales S (in thousands of dollars) that are attributed to an advertising campaign are given by
where t is the number of weeks the campaign runs.
(a) Find the rate of change of sales at any time t.
(b) Use the second derivative to find how this rate changing at t = 15.
(c) Interpret your result in part (b).
42. Advertising and sales A product with a large advertising budget has its sales S (in millions of dollars) given by
where t is the number of months the product has been on the market.
(a) Find the rate of change of sales at any time t.
(b) What is the rate of change of sales at t = 2?
(c) Use the second derivative to find how this rate is changing at t = 2.
(d) Interpret your results from parts (b) and (c).
43E44E47ESuppose the total cost function for a commodity is .
1. Find the marginal cost function.
Suppose the total cost function for a commodity is
2. What is the marginal cost if x = 50 units are produced?
Suppose the total cost function for a commodity is .
3. Use marginal cost to estimate the cost of producing the 51st unit.
Suppose the total cost function for a commodity is .
4. Calculate to find the actual cost of producing the 51st unit.
Suppose the total cost function for a commodity is .
5. True or false: For products that have linear cost functions, the actual cost of producing the st unit is equal to the marginal cost at x.
6CPIf the total profit function for a product is , then the marginal profit and its derivative are
7. Is the marginal profit decreasing for all ?
In Problems 1-8, total revenue is in dollars and x is the number of units.
1. (a) If the total revenue function for a product is , what is the marginal revenue function for that product?
(b) What does this marginal revenue function tell us?
In Problems 1-8, total revenue is in dollars and x is the number of units.
2. If the total revenue function for a product is , what is the marginal revenue for the product? What does this mean?
MARGINAL REVENUE, COST, AND PROFIT In Problems 1-8, total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is R=36x0.01x2. (a) Find R(100) and tell what it represents. (b) Find the marginal revenue function. (c) Find the marginal revenue at x = 100, and tell what it predicts about the sale of the next unit and the next 3 units. (d) Find R(101)R(100) and explain what this value represents.MARGINAL REVENUE, COST, AND PROFIT In Problems 1-8, total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is R(x)=25x0.05x2. (a) Find R(50) and tell what it represents. (b) Find the marginal revenue function. (c) Find the marginal revenue at x = 50, and tell what it predicts about the sale of the next unit and the next 2 units. (d) Find R(51)R(50) and explain what this value represents.5EIn Problems 1-8, total revenue is in dollars and x is the number of units.
6. Suppose that in a monopoly market, the demand function for a product is given by
where x is the number of units and p is the price in dollars.
(a) Find the total revenue from the sale of 500 units.
(b) Find and interpret the marginal revenue at 500 units.
(c) Is more revenue expected from the 501st unit sold or from the 701st? Explain.
MARGINAL REVENUE, COST, AND PROFIT In Problems 1-8, total revenue is in dollars and x is the number of units. (a) Graph the marginal revenue function from Problem 3. (b) At what value of x will total revenue be maximized for Problem 3. (c) What is the maximum revenue? Suppose that the total revenue function for a commodity is R=36x0.01x2. (a) Find R(100) and tell what it represents. (b) Find the marginal revenue function. (c) Find the marginal revenue at x = 100, and tell what it predicts about the sale of the next unit and the next 3 units. (d) Find R(101)R(100) and explain what this value represents.8E9E10EIn Problems 9-16, cost is in dollars and x is the number of units. Find the marginal cost functions for the given cost functions.
12E13E14EIn Problems 9-16, cost is in dollars and x is the number of units. Find the marginal cost functions for the given cost functions.
16E17E18E19. If the cost function for a commodity is
dollars
find the marginal cost at units and tell what this predicts about the cost of producing 1 additional unit and 3 additional units.
20E21E22EIn each of Problems 23 and 24, the graph of a company’s total cost function is shown. For each problem, use the graph to answer the following questions.
(a) Will the 101st item or the 501st item cost more to produce? Explain.
(b) Does this total cost function represent a manufacturing process that is getting more efficient or less efficient? Explain.
24E25E26EIn Problems 25-28, cost, revenue, and profit are in dollars and x is the number of units.
27. Suppose that the total revenue function for a product is and that the total cost function is .
(a) Find the profit from the production and sale of 500 units.
(b) Find the marginal profit function.
(c) Find and explain what it predicts.
(d) Find and explain what this value represents.
28EIn each of Problems 29 and 30, the graphs of a company’s total revenue function and total cost function are shown. For each problem, use the graph to answer the following questions.
(a) From the sale of 100 items, 400 items, and 700 items, rank from smallest to largest the amount of profit received. Explain your choices and note whether any of these scenarios results in a loss.
(b) From the sale of the 101st item, the 401st item, and the 701st item, rank from smallest to largest the amount of profit received. Explain your choices, and note whether any of these scenarios results in a loss.
In each of Problems 29 and 30, the graphs of a company’s total revenue function and total cost function are shown. For each problem, use the graph to answer the following questions.
(a) From the sale of 100 items, 400 items, and 700 items, rank from smallest to largest the amount of profit received. Explain your choices and note whether any of these scenarios results in a loss.
(b) From the sale of the 101st item, the 401st item, and the 701st item, rank from smallest to largest the amount of profit received. Explain your choices, and note whether any of these scenarios results in a loss.
31E32E33E34E35. The price of a product in a competitive market is $300. If the cost per unit of producing the product is dollars, where x is the number of units produced per month, how many units should the firm produce and sell to maximize its profit?
36. The cost per unit of producing a product is dollars, where x represents the number of units produced per week. If the equilibrium price determined by a competitive market is $220, how many units should the firm produce and sell each week to maximize its profit?
37. If the daily cost per unit of producing a product by the Ace Company is dollars and the price on the competitive market is $70, what is the maximum daily profit the Ace Company can expect on this product?
38. The Mary Ellen Candy Company produces chocolate Easter bunnies at a cost per unit of 0. dollars, where x is the number produced. If the price on the competitive market for a bunny this size is $10.00, how many should the company produce to maximize its profit?
In Problems 1-4, find all critical points and determine whether they are relative maxima, relative minima, or horizontal points of inflection.
2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE