Explanation Given: From the figure it is observed that the rectangular region R represents the financial district of the city. The price of the land within the district is approximately given in dollars as, p ( x , y ) = 200 − 10 ( x − 1 2 ) 2 − 15 ( y − 1 ) 2 (1) Here, x and y are measured in miles. p ( x , y ) represents the price of the land at the particular point ( x , y ) . Result used: “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. It is to be noted that in (a) and (b) f x x and f y y must have the same sign to satisfy the condition D ( a , b ) > 0 .” Calculation: The point at which the price of the land is high is obtained as follows. Obtain the critical point of the function P ( x , y ) . The first derivatives of the function are computed as follows, Partially differentiate the equation (1) with respect to x . ∂ ∂ x P ( x , y ) = ∂ ∂ x ( 200 − 10 ( x − 1 2 ) 2 − 15 ( y − 1 ) 2 ) P x = − 20 ( x − 1 2 ) (2) Partially differentiate the equation (1) with respect to y . ∂ ∂ y P ( x , y ) = ∂ ∂ y ( 200 − 10 ( x − 1 2 ) 2 − 15 ( y − 1 ) 2 ) P y = − 30 ( y − 1 ) (3) Substitute p x ( x , y ) = 0 and p y ( x , y ) = 0 in the equation (2) and equation (3), Then equation (2) becomes, − 20 ( x − 1 2 ) = 0 x = 1 2 Thus, the value of x is 1 2 . Equation (3) becomes, − 30 ( y − 1 ) = 0 y = 1 Thus, the value of y is 1. The critical point of the given function is ( 1 2 , 1 ) . Thus, price of the land is high at ( 1 2 , 1 )

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ISBN: 9781337825597

Author: Tan

Publisher: INTER CENG

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Chapter 8.3, Problem 23E

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To find: The point at which the price of the financial land is highest.

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Chapter 8.3, Problem 22E

Chapter 8.3, Problem 24E

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Chapter 8 Solutions

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The point at which the price of the financial land is highest.

Solution Summary: The author explains that the rectangular region R represents the financial district of the city.

To find: The point at which the price of the financial land is highest.

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Chapter 8 Solutions