Explanation Given: The function is, f ( x , y ) = e x ( y + 1 ) . 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 ) = e x ( y + 1 ) . Obtain f x ( x , y ) , and f y ( x , y ) as follows: f x ( x , y ) = ∂ ∂ x ( e x ( y + 1 ) ) = ( y + 1 ) e x ( y + 1 ) and, f y ( x , y ) = ∂ ∂ y ( e x ( y + 1 ) ) = x e x ( y + 1 ) The function has relative extrema at a point, if f x ( x , y ) = 0 ( y + 1 ) e x ( y + 1 ) = 0 ( y + 1 ) = 0 ( ∵ e x ( y + 1 ) ≠ 0 ) y = − 1 And if f y ( x , y ) = 0 , f y ( x , y ) = 0 x e x ( y + 1 ) = 0 x = 0 Therefore, ( 0 , 0 ) is the critical point where extrema can occur

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 9.3, Problem 17E

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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 16E

Chapter 9.3, Problem 18E

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Chapter 9 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

Ch. 9.1 - YOUR TURN For the function in Example 1, find...Ch. 9.1 - Prob. 2YT Ch. 9.1 - Prob. 3YT Ch. 9.1 - EXERCISES Let f(x,y)=2x3y+5.Find the following. a....Ch. 9.1 - Prob. 2E Ch. 9.1 - EXERCISES Let h(x,y)=x2+2y2. Find the following....Ch. 9.1 - EXERCISES Let f(x,y)=9x+5ylogx. Find the...Ch. 9.1 - EXERCISES Let f(x,y)=ex+ln(x+y). Find the...Ch. 9.1 - EXERCISES Let f(x,y)=xex+y. Find the following. a....Ch. 9.1 - EXERCISES Let f(x,y)=xsin(x2y). Find the...

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

Solution Summary: The author explains that the function has no relative extrema and the saddle point for the given function is (0,-1).

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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.