Explanation Given: The function is f ( x , y ) = ln ( 1 + x 2 + y 2 ) (1) Result used: Second derivative test “Let D ( x , y ) = f x x ( x , y ) f y y ( x , y ) − f 2 x y ( x , y ) . Then, (a) If D ( a , b ) > 0 and f x x < 0 then the function f has a relative maximum at the critical point ( a , b ) . (b) If D ( a , b ) > 0 and f x x > 0 then the function f has a relative minimum at the critical point ( a , b ) . (c) If D ( a , b ) < 0 then the function f has a no relative extrema at the critical point ( a , b ) . The point ( a , b , f ( a , b ) ) is said to be a saddle point. (d) If D ( a , b ) = 0 then there will be no result.” Calculation: Obtain the critical point of the function . The partial derivatives of the given function is computed as follows, Partially differentiate the equation (1) with respect to x . ∂ ∂ x f ( x , y ) = ∂ ∂ x ( ln ( 1 + x 2 + y 2 ) ) = 1 1 + x 2 + y 2 ( 2 x ) ∂ ∂ x f ( x , y ) = 2 x 1 + x 2 + y 2 (2) Partially differentiate the equation (1) with respect to y . ∂ ∂ y f ( x , y ) = ∂ ∂ y ( ln ( 1 + x 2 + y 2 ) ) = 1 1 + x 2 + y 2 ( 2 y ) ∂ ∂ y f ( x , y ) = 2 y 1 + x 2 + y 2 (3) Thus, the partial derivatives are, f x = 2 x 1 + x 2 + y 2 and f y = 2 y 1 + x 2 + y 2 . Find the critical points by solving the equation f x ( x , y ) = 0 and f y ( x , y ) = 0 . The equation (2) written as ( 2 x ) 1 + x 2 + y 2 = 0 . 2 x = 0 ( 1 + x 2 + y 2 ) x = 0 Thus, the value of x is 0. The equation (3) can be written as ( 2 y ) 1 + x 2 + y 2 = 0 . 2 y = 0 ( 1 + x 2 + y 2 ) y = 0 Thus, the value of y is 0. Therefore, the critical point of the given function is ( 0 , 0 ) . Partially differentiate the equation (2) with respect to x . ∂ 2 ∂ x 2 f ( x , y ) = ∂ ∂ x ( ( 2 x ) 1 + x 2 + y 2 ) (4) Use the mention formula ∂ u ∂ v = v ∂ ∂ u − u ∂ ∂ v v 2 , to obtain the value of ∂ 2 ∂ x 2 f ( x , y ) , Here v = 1 + x 2 + y 2 and u = 2 x . The equation (4) can be written as follows, ∂ 2 ∂ x 2 f ( x , y ) = ( 1 + x 2 + y 2 ) ( 2 ) − ( 2 x ) ( 2 x ) ( 1 + x 2 + y 2 ) 2 f x x = 2 + 2 x 2 + 2 y 2 − 4 x 2 ( 1 + x 2 + y 2 ) 2 = 2 ( 1 − x 2 + y 2 ) ( 1 + x 2 + y 2 ) 2 Partially differentiate the equation (3) with respect to y . ∂ 2 ∂ y 2 f ( x , y ) = ∂ ∂ y ( ( 2 y ) 1 + x 2 + y 2 ) (5) Use the mentioned formula ∂ u ∂ v = v ∂ ∂ u − u ∂ ∂ v v 2 and take the following value of u and v as taken below. Here v = 1 + x 2 + y 2 and u = 2 y . Then the equation (5) becomes as follows, ∂ 2 ∂ x 2 f ( x , y ) = ( 1 + x 2 + y 2 ) ( 2 ) − ( 2 y ) ( 2 y ) ( 1 + x 2 + y 2 ) 2 = 2 + 2 x 2 + 2 y 2 − 4 y 2 ( 1 + x 2 + y 2 ) 2 f y y = 2 ( 1 + x 2 − y 2 ) ( 1 + x 2 + y 2 ) 2 Similarly, partially differentiate the equation (2) with respect to y

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

10th Edition

ISBN: 9781285464640

Author: Soo T. Tan

Publisher: Brooks Cole

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Chapter 8.3, Problem 19E

To determine

The critical points and relative extrema of the function; Classify the nature of each point if possible.

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Chapter 8.3, Problem 18E

Chapter 8.3, Problem 20E

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - If f is a function of two variables, what can you...Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Let g(x, y) = 2x2 y2. Compute g(1, 2), g(2,1),...Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Let h(s, t) = s ln t t ln s. Compute h(1, e),...

Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 18E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 24E Ch. 8.1 - Find an equation of the level curve of f (x, y) =...Ch. 8.1 - Prob. 26E Ch. 8.1 - In Exercises 27 and 28, match the graph of the...Ch. 8.1 - Prob. 28E Ch. 8.1 - Can two level curves of a function f of two...Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - PRICE-TO-EARNINGS Ratio The current PE ratio...Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - REVENUE FUNCTION FOR ELECTRIC CARS Bell Motors...Ch. 8.1 - Prob. 39E Ch. 8.1 - REVENUE FUNCTION FOR DESKS For the total revenue...Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - SURFACE AREA OF A HUMAN BODY An empirical formula...Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - PRODUCTION FUNCTION FOR A COUNTRY Suppose the...Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - HOME MORTGAGES The monthly payment that amortizes...Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - Prob. 55E Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.1 - Prob. 61E Ch. 8.1 - Prob. 62E Ch. 8.1 - Prob. 63E Ch. 8.1 - Prob. 64E Ch. 8.2 - Prob. 1CQ Ch. 8.2 - Prob. 2CQ Ch. 8.2 - Prob. 3CQ Ch. 8.2 - Prob. 4CQ Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - In Exercises 3-24, find the first partial...Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - In Exercises 3-24, find the first partial...Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - Prob. 42E Ch. 8.2 - Prob. 43E Ch. 8.2 - Prob. 44E Ch. 8.2 - Prob. 45E Ch. 8.2 - Prob. 46E Ch. 8.2 - Land Prices The rectangular region R shown in the...Ch. 8.2 - Prob. 48E Ch. 8.2 - Prob. 49E Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Surface area of a human body The formula S =...Ch. 8.2 - Wind Chill Factor A formula used by meteorologists...Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.2 - Complementary and Substitute Commodities In a...Ch. 8.2 - Complementary and Substitute commodities In a...Ch. 8.2 - Prob. 59E Ch. 8.2 - Prob. 60E Ch. 8.2 - Prob. 61E Ch. 8.2 - Prob. 62E Ch. 8.2 - According to the ideal gas law, the volume V (in...Ch. 8.2 - Kinetic Energy of a Body The kinetic energy ODYK...Ch. 8.2 - Prob. 66E Ch. 8.2 - Prob. 67E Ch. 8.2 - Prob. 68E Ch. 8.2 - Prob. 69E Ch. 8.2 - Prob. 70E Ch. 8.2 - Prob. 72E Ch. 8.2 - Prob. 1TE Ch. 8.2 - Prob. 2TE Ch. 8.2 - Prob. 3TE Ch. 8.2 - Prob. 4TE Ch. 8.2 - Prob. 5TE Ch. 8.2 - Prob. 6TE Ch. 8.3 - Prob. 1CQ Ch. 8.3 - Prob. 2CQ Ch. 8.3 - Prob. 3CQ Ch. 8.3 - Prob. 4CQ Ch. 8.3 - In Exercises 1-20, find the critical point(s) of...Ch. 8.3 - Prob. 2E Ch. 8.3 - In Exercises 1-20, find the critical point(s) of...Ch. 8.3 - Prob. 4E Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - In Exercises 1-20, find the critical point(s) of...Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Maximization Profit The total weekly revenue (in...Ch. 8.3 - Maximizing Profit The total daily revenue (in...Ch. 8.3 - Prob. 23E Ch. 8.3 - Maximizing Profit CG Imports imports two brands of...Ch. 8.3 - Maximizing Revenue The management of Cal...Ch. 8.3 - Maximizing Profit Johnsons Household Products has...Ch. 8.3 - Maximizing Profit Johnsons Household Products has...Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Parcel Post Regulators Postal regulations specify...Ch. 8.3 - Packaging An open rectangular box having a volume...Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Prob. 40E Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.4 - Prob. 1CQ Ch. 8.4 - Prob. 2CQ Ch. 8.4 - Prob. 4CQ Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Facebook Users End-of-year data for the number of...Ch. 8.4 - E-Book Audience The number of adults (in millions)...Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8.4 - Prob. 27E Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - Prob. 1TE Ch. 8.4 - Prob. 2TE Ch. 8.4 - Prob. 3TE Ch. 8.4 - Prob. 4TE Ch. 8.4 - Prob. 5TE Ch. 8.4 - Prob. 6TE Ch. 8.4 - Prob. 7TE Ch. 8.4 - Prob. 8TE Ch. 8.4 - Prob. 9TE Ch. 8.5 - Prob. 1CQ Ch. 8.5 - Prob. 2CQ Ch. 8.5 - Prob. 1E Ch. 8.5 - In Exercises 1-16. use the method of Lagrange...Ch. 8.5 - Prob. 3E Ch. 8.5 - Prob. 4E Ch. 8.5 - Prob. 5E Ch. 8.5 - Prob. 6E Ch. 8.5 - Prob. 7E Ch. 8.5 - Prob. 8E Ch. 8.5 - Prob. 9E Ch. 8.5 - Prob. 10E Ch. 8.5 - Prob. 11E Ch. 8.5 - Prob. 12E Ch. 8.5 - Prob. 13E Ch. 8.5 - Prob. 14E Ch. 8.5 - In Exercises 1-16. use the method of Lagrange...Ch. 8.5 - Prob. 16E Ch. 8.5 - Prob. 17E Ch. 8.5 - Maximizing Profit The total daily profit (in...Ch. 8.5 - Minimizing Construction Costs The management of...Ch. 8.5 - Prob. 20E Ch. 8.5 - Packaging Find the dimensions of an open...Ch. 8.5 - Maximizing Profit The Ace Novelty company produces...Ch. 8.5 - Prob. 23E Ch. 8.5 - Parcel Post Regulations Postal regulations specify...Ch. 8.5 - Minimizing Container Costs The Betty Moore Company...Ch. 8.5 - Minimizing Construction Costs An open rectangular...Ch. 8.5 - Prob. 27E Ch. 8.5 - Prob. 28E Ch. 8.5 - Prob. 29E Ch. 8.5 - Prob. 30E Ch. 8.5 - Maximizing Production Suppose a company can...Ch. 8.5 - Prob. 33E Ch. 8.5 - Prob. 34E Ch. 8.5 - Prob. 36E Ch. 8.5 - Prob. 38E Ch. 8.5 - In Exercises 39-42, determine whether the...Ch. 8.5 - Prob. 40E Ch. 8.5 - Prob. 41E Ch. 8.5 - Prob. 42E Ch. 8.6 - Prob. 1CQ Ch. 8.6 - Prob. 2CQ Ch. 8.6 - Prob. 3CQ Ch. 8.6 - Prob. 4CQ Ch. 8.6 - Prob. 5CQ Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - In Exercises 1-25. evaluate the double integral...Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - Prob. 15E Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - In Exercises 26-33, use a double integral to find...Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Prob. 37E Ch. 8.6 - Prob. 38E Ch. 8.6 - Prob. 39E Ch. 8.6 - Prob. 40E Ch. 8.6 - Prob. 41E Ch. 8.6 - Prob. 42E Ch. 8.6 - Prob. 43E Ch. 8.6 - In Exercises 42-47, find the average value of the...Ch. 8.6 - Prob. 45E Ch. 8.6 - Prob. 46E Ch. 8.6 - Prob. 47E Ch. 8.6 - Prob. 48E Ch. 8.6 - Prob. 49E Ch. 8.6 - Population Density Of A City The population...Ch. 8.6 - Prob. 51E Ch. 8.6 - Prob. 52E Ch. 8.6 - Prob. 53E Ch. 8.6 - Prob. 54E Ch. 8.6 - Prob. 55E Ch. 8.6 - Prob. 56E Ch. 8 - Fill in the blanks. 1. The domain of a function f...Ch. 8 - Prob. 2CRQ Ch. 8 - Prob. 3CRQ Ch. 8 - Prob. 4CRQ Ch. 8 - Fill in the blanks. 5. The partial derivative fx...Ch. 8 - Prob. 6CRQ Ch. 8 - Prob. 7CRQ Ch. 8 - Prob. 8CRQ Ch. 8 - Prob. 9CRQ Ch. 8 - Prob. 10CRQ Ch. 8 - Prob. 11CRQ Ch. 8 - Prob. 12CRQ Ch. 8 - Prob. 1RE Ch. 8 - Prob. 2RE Ch. 8 - Prob. 3RE Ch. 8 - Prob. 4RE Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE Ch. 8 - Prob. 17RE Ch. 8 - Prob. 18RE Ch. 8 - Prob. 19RE Ch. 8 - Prob. 20RE Ch. 8 - Prob. 21RE Ch. 8 - Prob. 22RE Ch. 8 - Prob. 23RE Ch. 8 - Prob. 24RE Ch. 8 - Prob. 25RE Ch. 8 - Prob. 26RE Ch. 8 - Prob. 27RE Ch. 8 - Prob. 28RE Ch. 8 - Prob. 29RE Ch. 8 - Prob. 30RE Ch. 8 - Prob. 31RE Ch. 8 - Prob. 32RE Ch. 8 - Prob. 33RE Ch. 8 - Prob. 34RE Ch. 8 - Prob. 35RE Ch. 8 - Prob. 36RE Ch. 8 - Prob. 37RE Ch. 8 - Prob. 38RE Ch. 8 - Prob. 39RE Ch. 8 - Prob. 40RE Ch. 8 - Prob. 41RE Ch. 8 - Prob. 42RE Ch. 8 - Prob. 43RE Ch. 8 - Prob. 44RE Ch. 8 - Prob. 45RE Ch. 8 - Prob. 46RE Ch. 8 - Revenue Functions A division of Ditton Industries...Ch. 8 - Prob. 48RE Ch. 8 - Prob. 49RE Ch. 8 - Prob. 50RE Ch. 8 - Prob. 51RE Ch. 8 - Maximizing Revenue Odyssey Travel Agencys monthly...Ch. 8 - Prob. 54RE Ch. 8 - Prob. 55RE Ch. 8 - Prob. 56RE Ch. 8 - Prob. 1BM Ch. 8 - Prob. 2BM Ch. 8 - Prob. 3BM Ch. 8 - Prob. 4BM Ch. 8 - Prob. 5BM Ch. 8 - Evaluate R(1xy)dA, where R is the region bounded...

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The critical points and relative extrema of the function; Classify the nature of each point if possible.

Solution Summary: The author explains how to determine the critical points and relative extrema of the function.

The critical points and relative extrema of the function; Classify the nature of each point if possible.

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Chapter 8 Solutions