Explanation Given: The given function is: z = − x 4 + y 4 + 2 x 2 − 2 y 2 + 1 16 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, z = − x 4 + y 4 + 2 x 2 − 2 y 2 + 1 16 Find the f x ( x , y ) , and f y ( x , y ) : f x ( x , y ) = ∂ ∂ x ( − x 4 + y 4 + 2 x 2 − 2 y 2 + 1 16 ) = − 4 x 3 + 4 x and, f y ( x , y ) = ∂ ∂ y ( − x 4 + y 4 + 2 x 2 − 2 y 2 + 1 16 ) = 4 y 3 − 4 y A function has relative extrema at a point, if f x ( x , y ) = 0 − 4 x 3 + 4 x = 0 − 4 x ( x 2 − 1 ) = 0 x = 0 , ± 1 And if f y ( x , y ) = 0 , f y ( x , y ) = 0 4 y 3 − 4 y = 0 4 y ( y 2 − 1 ) = 0 y = 0 , ± 1 Therefore, ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , − 1 ) , ( − 1 , 0 ) , ( 1 , 0 ) , ( 1 , − 1 ) , ( 1 , 1 ) , ( − 1 , 1 ) and ( 1 , 1 ) are the critical point where extrema can occur

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 25E

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The all point for relative extrema and then match the equation to its graph for the given function.

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Chapter 9.3, Problem 24E

Chapter 9.3, Problem 26E

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The all point for relative extrema and then match the equation to its graph for the given function.

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To find: The all point for relative extrema and then match the equation to its graph for the given function.

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Chapter 9 Solutions

To find:

The all point for relative extrema and then match the equation to its graph for the given function.