Explanation From the given statement it is observed that ( a , b ) is a critical point of f and the value of f x x ( a , b ) and f y y ( a , b ) are negative. Apply the given conditions in the second derivative test, D ( x , y ) = f x x ( x , y ) f y y ( x , y ) − ( f x y ( x , y ) ) 2 Since f x x ( a , b ) and f y y ( a , b ) are negative, f x x ( a , b ) f y y ( a , b ) is positive. Similarly, ( f x y ( x , y ) ) 2 is positive and it will be greater than f x x ( a , b ) f y y ( a , b ) . Thus, the second derivative test for the above condition becomes, D ( a , b ) < 0 . Recall the condition “If D ( a , b ) > 0 and f x x ( a , b ) < 0 , then the function has a relative maximum.” in the second derivative test D ( x , y ) = f x x ( x , y ) f y y ( x , y ) − ( f x y ( x , y ) ) 2 . From the above condition, it is noted that the D ( a , b ) < 0 does not satisfy the condition of second derivative test to be a relative maximum. Hence, the statement is false. Example: Let us consider a function f ( x , y ) = − x 2 − y 2 + 3 x y (1) The derivatives of the function are computed as follows. The first derivatives of f ( x , y ) = − x 2 − y 2 + 3 x y are, Partially differentiate the equation (1) with respect to x , ∂ ∂ x ( f ( x , y ) ) = ∂ ∂ x ( − x 2 − y 2 + 3 x y ) f x ( x , y ) = − 2 x + 3 y (2) Partially differentiate the equation (1) with respect to y , ∂ ∂ y ( f ( x , y ) ) = ∂ ∂ y ( − x 2 − y 2 + 3 x y ) f y ( x , y ) = − 2 y + 3 x (3) Thus, the partial derivatives are f x = − 2 x + 3 y and f y = − 2 y + 3 x . Find the critical points by assuming f x ( x , y ) = 0 and f y ( x , y ) = 0 . Then equation (2) becomes, − 2 x + 3 y = 0 (4) Isolate the value of x from the above equation, − 2 x = − 3 y x = 3 2 y And the equation (4) becomes, − 2 y + 3 x = 0 (5) Substitute x = 3 2 y in the equation (5), − 2 y + 3 ( 3 2 y ) = 0 − 2 y + − 9 2 y = 0 y = 0 Thus the value of y is 0

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 37E

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Whether the statement “If (a,b) is a critical point of f and both the conditions fxx(a,b)<0 and fyy(a,b)<0 hold, then f has a relative maximum” is true or false. Explain the reason if it is true or given an example if it is false.

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Chapter 8.3, Problem 36E

Chapter 8.3, Problem 38E

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Chapter 8 Solutions

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

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Whether the statement “If ( a , b ) is a critical point of f and both the conditions f x x ( a , b ) < 0 and f y y ( a , b ) < 0 hold, then f has a relative maximum ” is true or false. Explain the reason if it is true or given an example if it is false.

Solution Summary: The author explains whether the statement "If (a,b) is a critical point of f and both the conditions

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Whether the statement “If (a,b) is a critical point of f and both the conditions fxx(a,b)<0 and fyy(a,b)<0 hold, then f has a relative maximum” is true or false. Explain the reason if it is true or given an example if it is false.

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Chapter 8 Solutions

Whether the statement “If ( a , b ) is a critical point of f and both the conditions f x x ( a , b ) &lt; 0 and f y y ( a , b ) &lt; 0 hold, then f has a relative maximum ” is true or false. Explain the reason if it is true or given an example if it is false.

Solution Summary: The author explains whether the statement "If (a,b) is a critical point of f and both the conditions

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Whether the statement “If (a,b) is a critical point of f and both the conditions fxx(a,b)<0 and fyy(a,b)<0 hold, then f has a relative maximum” is true or false. Explain the reason if it is true or given an example if it is false.

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Chapter 8 Solutions

Whether the statement “If ( a , b ) is a critical point of f and both the conditions f x x ( a , b ) < 0 and f y y ( a , b ) < 0 hold, then f has a relative maximum ” is true or false. Explain the reason if it is true or given an example if it is false.