Explanation Given information: The function is f ( x , y ) = y e x − 3 x − y + 5 and the point is ( 0 , 3 ) . Explanation: Use the Second-Derivative test to determine the nature of the f ( x , y ) = y e x − 3 x − y + 5 at the point ( 0 , 3 ) . The Second-Derivative test: Suppose that f ( x , y ) is a function and ( a , b ) is a point at which ∂ f ∂ x ( a , b ) = 0 and ∂ f ∂ y ( a , b ) = 0 and D ( x , y ) = ∂ 2 f ∂ x 2 × ∂ 2 f ∂ y 2 − ( ∂ 2 f ∂ x ∂ y ) 2 There are four conditions to determine the relative maximum or minimum as follows: If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) > 0 , then f ( x , y ) has a relative minimum at ( a , b ) . If D ( a , b ) > 0 and ∂ 2 f ∂ x 2 ( a , b ) < 0 , then f ( x , y ) has a relative maximum at ( a , b ) . If D ( a , b ) < 0 , then f ( x , y ) has neither a relative maximum nor a relative minimum at ( a , b ) If D ( a , b ) = 0 , then no conclusion can be drawn from this test. Differentiate f ( x , y ) = y e x − 3 x − y + 5 partially twice with respect to x and y , it becomes ∂ f ∂ x = ∂ ∂ x ( y e x − 3 x − y + 5 ) ⇒ ∂ f ∂ x = y e x − 3 ∂ ∂ x ( ∂ f ∂ x ) = ∂ ∂ x ( y e x − 3 ) ⇒ ∂ 2 f ∂ x 2 = y e x

12th Edition

ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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Question

Chapter 7.3, Problem 23E

To determine

The nature of f(x, y)=yex−3x−y+5 at the point (0, 3) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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Chapter 7.3, Problem 22E

Chapter 7.3, Problem 24E

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The nature of f ( x , y ) = y e x − 3 x − y + 5 at the point ( 0 , 3 ) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

Solution Summary: The author explains the second-derivative test to determine the nature of f(x,y)=yex-3x-y+5 at the point (a,b

The nature of f(x, y)=yex−3x−y+5 at the point (0, 3) by using second-derivative test. If the second-derivative test is inconclusive, then state it.

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