Explanation Given: The function is f ( x , y ) = e 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 ( a , b ) < 0 implies that the function f has a relative maximum at the critical point ( a , b ) . (b) If D ( a , b ) > 0 and f x x ( a , b ) > 0 implies that the function f has a relative minimum at the critical point ( a , b ) . (c) If D ( a , b ) < 0 implies that 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 implies that the test is inconclusive, so some other technique must be used to solve the problem. Calculation: Obtain the critical point of the function f ( x , y ) = e x 2 + y 2 as follows. Compute the partial derivative of the equation (1) with respect to x . ∂ ∂ x f ( x , y ) = ∂ ∂ x ( e x 2 + y 2 ) ∂ ∂ x f ( x , y ) = e x 2 + y 2 ( 2 x ) (2) Compute the partial derivative of the equation (1) with respect to y . ∂ ∂ y f ( x , y ) = ∂ ∂ y ( e x 2 + y 2 ) f ( x , y ) = e x 2 + y 2 ( 2 y ) (3) Thus, the partial derivatives are f x = e x 2 + y 2 ( 2 x ) and f y = e x 2 + y 2 ( 2 y ) . Obtain the critical points by equating f x ( x , y ) and f y ( x , y ) equal to zero. The equation (2) becomes, e x 2 + y 2 ( 2 x ) = 0 2 x = 0 x = 0 Thus, the value of x is x = 0 . And equation (3) becomes, e x 2 + y 2 ( 2 y ) = 0 2 y = 0 y = 0 Thus, the value of y is y = 0 . Therefore, the critical point of the given function is ( 0 , 0 ) . Apply the second derivative test. Partially differentiate the equation (2) with respect to x . ∂ 2 ∂ x 2 f ( x , y ) = ∂ ∂ x ( e x 2 + y 2 ( 2 x ) ) f x x = ( ( 2 x ) e x 2 + y 2 ( 2 x ) + e x 2 + y 2 ( 2 ) ) = ( 4 e x 2 + y 2 ( x 2 ) + e x 2 + y 2 ( 2 ) ) f x x = 2 ( 2 e x 2 + y 2 ( x 2 ) + e x 2 + y 2 ) Partially differentiate the equation (2) with respect to y , ∂ 2 ∂ x y f ( x , y ) = ∂ ∂ y ( e x 2 + y 2 ( 2 x ) ) = e x 2 + y 2 ( 2 x ) ( 2 y ) f x y = 4 e x 2 + y 2 ( x y ) Partially differentiate the equation (3) with respect to y , ∂ 2 ∂ y 2 f ( x , 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 17E

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To find: 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 16E

Chapter 8.3, Problem 18E

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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 - 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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 - 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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 the critical points and relative extrema of the function.

To find: The critical points and relative extrema of the function; classify the nature of each point if possible.

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Chapter 8 Solutions