To determine Calculate the local maxima, minima , and saddle point for the function. Explanation Given information: The function is f ( x , y ) = e − y ( x 2 + y 2 ) Critical point: An interior point of the domain of a function f ( x , y ) where both f x and f y are zero or where one or both or where one or both of f x and f y do not exist is a critical point of f . Theorem:11 Suppose that f ( x , y ) and its first and second partial derivative is continuous throughout a disk centered at ( a , b ) and that f x ( a , b ) = f y ( a , b ) = 0 . i. f has a local maximum at ( a , b ) if f x x < 0 and f x x f y y − f x y 2 > 0 at ( a , b ) . ii. f has a local minimum at ( a , b ) if f x x > 0 and f x x f y y − f x y 2 > 0 at ( a , b ) . iii. f has a saddle point at ( a , b ) if f x x f y y − f x y 2 < 0 at ( a , b ) . Calculation: Provide the function as shown below: f ( x , y ) = e − y ( x 2 + y 2 ) (1) Differentiate Equation (1) with respect to x . ∂ f ∂ x = ∂ ∂ x ( e − y ( x 2 + y 2 ) ) = ( 2 x + 0 ) e − y = 2 x e − y (2) Differentiate Equation (1) with respect to y. ∂ f ∂ y = ∂ ∂ y ( e − y ( x 2 + y 2 ) ) = ∂ ∂ y ( y 2 + x 2 ) e − y + ( y 2 + x 2 ) ∂ ∂ y ( e − y ) = ( ∂ ∂ y ( y 2 ) + ∂ ∂ y ( x 2 ) ) e − y + e − y ∂ ∂ y ( − y ) ( y 2 + x 2 ) = ( − ∂ ∂ y ( y ) ( y 2 + x 2 ) e − y ) + ( 2 y + 0 ) e − y = 2 y e − y − 1 ( y 2 + x 2 ) e − y = 2 y e − y − e − y ( x 2 + y 2 ) (3) Find the critical point as shown below: Equate first order partial derivative with respect to x ∂ f ∂ x = 0 . 2 x e − y = 0 (4) Equate first order partial derivative with respect to y ∂ f ∂ y = 0 . 2 y e − y − e − y ( x 2 + y 2 ) = 0 (5) Solving the Equation (4) and (5) to get values 2 x e − y = 0 2 y e − y − e − y ( x 2 + y 2 ) = 0 Thus, the critical points are ( 0 , 0 ) and ( 0 , 2 ) . Differentiate Equation (2) with respect to x at ( 0 , 0 ) . ∂ 2 f ∂ x 2 = ∂ ∂ x ( 2 x e − y ) = 2 e − y (6) Substitute 0 for y

Thomas' Calculus: Early Transcendentals, Single Variable (14th Edition)

14th Edition

ISBN: 9780134439419

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14.7, Problem 27E

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Calculate the local maxima, minima, and saddle point for the function.

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Chapter 14.7, Problem 26E

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Thomas' Calculus: Early Transcendentals, Single Variable (14th Edition)

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