Explanation 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 y + x 3 − y 3 + 1 8 Find f x ( x , y ) , and f y ( x , y ) : f x ( x , y ) = ∂ ∂ x ( − 3 x y + x 3 − y 3 + 1 8 ) = − 3 y + 3 x 2 and, f y ( x , y ) = ∂ ∂ y ( − 3 x y + x 3 − y 3 + 1 8 ) = − 3 x − 3 y 2 A function has relative extrema at a point, if f x ( x , y ) = 0 − 3 y + 3 x 2 = 0 And if f y ( x , y ) = 0 , f y ( x , y ) = 0 − 3 x − 3 y 2 = 0 Solve the equation − 3 y + 3 x 2 = 0 and − 3 x − 3 y 2 = 0 using substitution, we get: x = 0 , y = 0 and x = − 1 , y = 1 . Therefore, ( 0 , 0 ) and ( − 1 , 1 ) are the critical point where extrema can occur. Check for saddle point:

2nd Edition

ISBN: 9780321964458

Author: Lial

Publisher: PEARSON EDUCATION (COLLEGE)

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Chapter 9.3, Problem 21E

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All relative extrema points for the function z=−3xy+x3−y3+18 and then match the equation to its graph for the given function.

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Chapter 9.3, Problem 20E

Chapter 9.3, Problem 22E

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All relative extrema points for the function z = − 3 x y + x 3 − y 3 + 1 8 and then match the equation to its graph for the given function.

Solution Summary: The author explains how to find all relative extrema points for the function z=-3xy+x3-y3, plus 18 and match the equation to its

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To find:

All relative extrema points for the function z=−3xy+x3−y3+18 and then match the equation to its graph for the given function.

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Chapter 9 Solutions

All relative extrema points for the function z = − 3 x y + x 3 − y 3 + 1 8 and then match the equation to its graph for the given function.

Solution Summary: The author explains how to find all relative extrema points for the function z=-3xy+x3-y3, plus 18 and match the equation to its

Concept explainers

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To find: All relative extrema points for the function z=−3xy+x3−y3+18 and then match the equation to its graph for the given function.

Want to see the full answer?

Chapter 9 Solutions

To find:

All relative extrema points for the function z=−3xy+x3−y3+18 and then match the equation to its graph for the given function.