Explanation Result used: Second Derivative Test: “Suppose the second partial derivatives of f are continuous on a disk with center ( a , b ) , and suppose that f x ( a , b ) = 0 and f y ( a , b ) = 0 (that is ( a , b ) is a critical point of f ). Let D ( a , b ) = f x x ( a , b ) f y y ( a , b ) − [ f x y ( a , b ) ] 2 ( a ) If D ( a , b ) > 0 and f x x ( a , b ) > 0 , then f ( a , b ) is a local minimum . ( b ) If D ( a , b ) > 0 and f x x ( a , b ) < 0 , then f ( a , b ) is a local maximum . ( c ) If D ( a , b ) < 0 , then f ( a , b ) is a saddle point. ( d ) If D ( a , b ) = 0 , then the test is inconclusive”. Calculation: The given function is, f ( x , y ) = x 4 y 2 . Take the partial derivative in the given function with respect to x and obtain f x . ∂ f ∂ x = ∂ ∂ x ( x 4 y 2 ) = y 2 ∂ ∂ x ( x 4 ) = y 2 ( 4 x 3 ) = 4 x 3 y 2 Thus, ∂ f ∂ x = 4 x 3 y 2 (1) Take the partial derivative in the given function with respect to y and obtain f y . ∂ f ∂ y = ∂ ∂ y ( x 4 y 2 ) = x 4 ∂ ∂ y ( y 2 ) = x 4 ( 2 y ) = 2 x 4 y Thus, ∂ f ∂ y = 2 x 4 y (2) Set the above partial derivatives to 0 and find the values of x and y . From the equation (1), 4 x 3 y 2 = 0 x 3 y 2 = 0 If x = 0 is fixed then y can take any value. From the equation (2), 2 x 4 y = 0 x 4 y = 0 If y = 0 is fixed then x can take any value

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 12.8, Problem 41E

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To show: The second derivative test is inconclusive for the function f(x,y)=x4y2 at (0,0) ; explain the behavior of the function at the critical point.

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Chapter 12.8, Problem 40E

Chapter 12.8, Problem 42E

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The second derivative test is inconclusive for the function f ( x , y ) = x 4 y 2 at ( 0 , 0 ) ; explain the behavior of the function at the critical point.

Solution Summary: The author explains that the second derivative test is inconclusive for the function f(x,y)=x4y

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To show: The second derivative test is inconclusive for the function f(x,y)=x4y2 at (0,0) ; explain the behavior of the function at the critical point.

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Chapter 12 Solutions