Explanation Given: The function is, f ( x , y ) = 3 x 2 + 7 y 3 − 42 x y + 5 . Approach: “For a function z = f ( x , y ) , let f x x , f y y , and f x y all exist in a circular region contained in the xy -plane with center ( a , b ) . Further, let f x ( a . b ) = 0 and f y ( a . b ) = 0 . Define the number D , known as the discriminant, by D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 Then, a. f ( a , b ) is a relative maximum if D > 0 and f x x ( a , b ) < 0 . b. f ( a , b ) is a relative minimum if D > 0 and f x x ( a , b ) > 0 . c. f ( a , b ) is a saddle point (neither a maximum nor a minimum) if D < 0 . d. if D = 0 , the test gives no information.” Calculation: Consider the given function, f ( x , y ) = 3 x 2 + 7 y 3 − 42 x y + 5 . Obtain f x ( x , y ) , and f y ( x , y ) as follows: f x ( x , y ) = ∂ ∂ x ( 3 x 2 + 7 y 3 − 42 x y + 5 ) = 6 x − 42 y and, f y ( x , y ) = ∂ ∂ y ( 3 x 2 + 7 y 3 − 42 x y + 5 ) = 21 y 2 − 42 x The function has relative extrema at a point, if f x ( x , y ) = 0 6 x − 42 y = 0 And if f y ( x , y ) = 0 , f y ( x , y ) = 0 21 y 2 − 42 x = 0 Solve the equation 6 x − 42 y = 0 and 21 y 2 − 42 x = 0 using substitution, 21 y 2 − 42 x = 0 21 y 2 − 42 ( 7 y ) = 0 ( ∵ 6 x − 42 y = 0 ) 21 y 2 − 294 y = 0 y = 0 or y = 14 Thus, the solutions are x = 0 , y = 0 and x = 98 , y = 14

2nd Edition

ISBN: 9780321964038

Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher: Pearson Addison Wesley,

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Chapter 9.3, Problem 16E

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The points for relative extrema and identify any saddle point for the given function.

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Chapter 9.3, Problem 15E

Chapter 9.3, Problem 17E

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The points for relative extrema and identify any saddle point for the given function.

Solution Summary: The author explains how to find the points for relative extrema and identify any saddle point for the given function.

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To find: The points for relative extrema and identify any saddle point for the given function.

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Chapter 9 Solutions

To find:

The points for relative extrema and identify any saddle point for the given function.