Use the derivative to find the vertex of the parabola.y =

Problem 1E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 2E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 3E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 4E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 57A Height After a great deal of experimentation, two Atlantic Institute of Technology senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to s(t) = ?16t2 + 40t + 3, where s is its height (in feet) above the ground t seconds after being released. a. How high will it go? ________________ b. How long is it in the air?

Problem 5E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 6E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 7E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 8E Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 9E For each of the exercise listed below, suppose that the function that is graphed is not f(x) but f?(x). Find the location of relative extrema, and tell whether each extremum is a relative maximum or minimum. Exercise Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 10E For each of the exercise listed below, suppose that the function that is graphed is not f(x) but f?(x). Find the location of relative extrema, and tell whether each extremum is a relative maximum or minimum. Exercise Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 12E For each of the exercise listed below, suppose that the function that is graphed is not f(x) but f?(x). Find the location of relative extrema, and tell whether each extremum is a relative maximum or minimum. Exercise Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 13E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x2 ? 10x + 33

Problem 11E For each of the exercise listed below, suppose that the function that is graphed is not f(x) but f?(x). Find the location of relative extrema, and tell whether each extremum is a relative maximum or minimum. Exercise Find the locations and values of all relative extrema for the functions with graphs as follows. Compare with Exercise in the preceding section.

Problem 14E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x2 + 8x + 5

Problem 15E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x3+ 6x2 + 9x ? 8

Problem 16E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x3 + 3x2 ? 24x + 2

Problem 17E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 18E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 20E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x4 ? 8x2 + 9

Problem 21E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = 3 ? (8 + 3x)2/3

Problem 22E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 19E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x4 ? 18x2 ? 4

Problem 23E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = 2x + 3x2/3

Problem 24E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = 3x5/3 ? 15x2/3

Problem 25E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 27E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 26E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 30E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = 3xex + 2

Problem 28E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 29E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x2ex ? 3

Problem 31E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = 2x + ln x

Problem 32E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 33E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema.

Problem 34E Find the x-value of all points where the function defined as follow have any relative extrema. Find the value of any relative extrema. f(x) = x + 8?x

Problem 36E Use the derivative to find the vertex of the parabola. y = ax2+ bx + c

Problem 35E Use the derivative to find the vertex of the parabola. y = ?2x2 + 12x ? 5

Problem 37E Graph the function on a graphing calculator, and then use the graph to find all relative extrema (to three decimal places). Then confirm your answer by finding the derivative and using the calculator to solve the equation f?(x) = 0. f(x) = x5?x4 + 4x3 ? 30x2 + 5x + 6

Problem 38E Graph the function on a graphing calculator, and then use the graph to find all relative extrema (to three decimal places). Then confirm your answer by finding the derivative and using the calculator to solve the equation f?(x) = 0. f(x) = ?x5?x4 + 2x3 ? 25x2 + 9x + 12

Problem 39E Graph f(x) = 2|x + 1| + 4|x ? 5|?20 with a graphing calculator in the window [?10, 10] by [?15, 30]. Use the graph and the function to determine the x-values of all extrema.

Problem 42A Profit In Exercise, find (a) the number, q, of units that produces maximum profit; (b) the price, p, per unit that produces maximum profit; and (c) the maximum profit, P. C(q) = 25q + 5000; p = 90 ? 0.02q

Problem 43A Profit In Exercise, find (a) the number, q, of units that produces maximum profit; (b) the price, p, per unit that produces maximum profit; and (c) the maximum profit, P. C(q) = 100 + 20qe?0.01q;p = 40e?0.01q

Problem 41A Profit In Exercise, find (a) the number, q, of units that produces maximum profit; (b) the price, p, per unit that produces maximum profit; and (c) the maximum profit, P. C(q) = 80 + 18q;p = 70 ? 2q

Problem 40E Consider the function . Source: Mathematics Teacher. a. Using a graphing calculator, try to find any local minima, or tell why finding a local minimum is difficult for this function. ________________ b. Find any local minima using the techniques of calculus. ________________ c. Based on your results in parts a and b, describe circumstances under which relative extrema are easier to find using the techniques of calculus than using a graphing calculator.

Problem 45A Power On August 8, 2007, the power used in New York state (in thousands of megawatts) could be approximated by the function P(t) = ?0.005846t3 + 0.1614t2 ? 0.4910t + 20.47, where t is the number of hours since midnight, for 0 ? t ? 24. Find any relative extrema for power usage, as well as when they occurred. Source: Current Energy.

Problem 46A Profit The total profit P(x) (in thousands of dollars) from the sale of x units of a certain prescription drug is given by P(x) = ln(?x3 + 3x2 + 72x + 1) for x in [0, 10]. a. Find the number of units that should be sold in order to maximize the total profit. ________________ b. What is the maximum profit?

Problem 44A Profit In Exercise, find (a) the number, q, of units that produces maximum profit; (b) the price, p, per unit that produces maximum profit; and (c) the maximum profit, P. C(q) = 21.047q + 3;p = 50 ? 5 ln(q + 10)

Problem 47A Revenue The demand equation for telephones at one store is p = D(q) = 200e?0.1q, where p is the price (in dollars) and q is the quantity of telephones sold per week. Find the values of q and p that maximize revenue.

Problem 48A Revenue The demand equation for one type of computer networking system is where p is the price (in dollars) and q is the quantity of servers sold per month. Find the values of q and p that maximize revenue.

Problem 50A Unemployment The annual unemployment rates of the U.S. civilian noninstitutional population for 1990–2009 are shown in the graph. Identify the years where relative extrema occur, and estimate the unemployment rate at each of these years. Source: Bureau of Labor Statistics.

Problem 49A Cost Suppose that the cost function for a product is given by C(x) = 0.002x3 + 9x + 6912. Find the production level (i.e., value of x) that will produce the minimum average cost per unit .

Problem 52A Milk Consumption The average individual daily milk consumption for herds of Charolais, Angus, and Hereford calves can be described by the function M(t) = 6.281t0.242e?0.025t, 1 ? t ? 26, where M(t) is the milk consumption (in kilograms) and t is the age of the calf (in weeks). Source: Animal Production. a. Find the time in which the maximum daily consumption occurs and the maximum daily consumption. ________________ b. If the general formula for this model is given by M(t) = atb e?ct, find the time where the maximum consumption occurs and the maximum consumption. (Hint:Express your answer in terms of a, b, and c.)

Problem 53A Alaskan Moose The mathematical relationship between the age of a captive female moose and its mass can be described by the function M(t) = 369(0.93)tt0.36, t ? 26, where M(t) is the mass of the moose (in kilograms) and t is the age (in years) of the moose. Find the age at which the mass of a female moose is maximized. What is the maximum mass? Source: Journal of Wildlife Management.

Problem 55A Attitude Change Social psychologists have found that as the discrepancy between the views of a speaker and those of an audience increases, the attitude change in the audience also increases to a point but decreases when the discrepancy becomes too large, particularly if the communicator is viewed by the audience as having low credibility. Suppose that the degree of change can be approximated by the function D(x) = –x4 + 8x3 + 80x2, where x is the discrepancy between the views of the speaker and those of the audience, as measured by scores on a questionnaire. Find the amount of discrepancy the speaker should aim for to maximize the attitude change in the audience. Source: Journal of Personality and Social Psychology.

Problem 51A Activity Level In the summer the activity level of a certain type of lizard varies according to the time of day. A biologist has determined that the activity level is given by the function a(t) = 0.008t3 2 0.288t2 + 2.304t + 7, where t is the number of hours after 12 noon. When is the activity level highest? When is it lowest?

Problem 54A Thermic Effect of Food As we saw in the last section, the metabolic rate after a person eats a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food and can be described for a particular individual as F(t) = ?10.28 + 175.9te?t/1.3, where F(t) is the thermic effect of food (in kJ/hr), and t is the number of hours that have elapsed since eating a meal. Find the time after the meal when the thermic effect of the food is maximized. Source: American Journal of Clinical Nutrition.

Problem 56A Film Length A group of researchers found that people prefer training films of moderate length; shorter films contain too little information, while longer films are boring. For a training film on the care of exotic birds, the researchers determined that the ratings people gave for the film could be approximated by where t is the length of the film (in minutes). Find the film length that received the highest rating.