Explanation Given: Function f ( x , y ) = y 3 − 3 x y + 6 x . Formula Used: Critical points of the function are the points where the function is defined and its derivative is undefined or zero. Mathematically, f x ( x 0 , y 0 ) = 0 f y ( x 0 , y 0 ) = 0 Let us assume that ( x 0 , y 0 ) is the critical point of a continuous function f and D be calculated as follows: D = f x x ( x 0 , y 0 ) f y y ( x 0 , y 0 ) − f x y ( x 0 , y 0 ) 2 Then, If D > 0 and f x x ( x 0 , y 0 ) > 0 , then the function has local minimum at ( x 0 , y 0 ) If D > 0 and f x x ( x 0 , y 0 ) < 0 , then the function has local maximum at ( x 0 , y 0 ) If D < 0 , then the function has saddle point at ( x 0 , y 0 ) If D = 0 , then test is inconclusive. Calculation: Function is given as f ( x , y ) = y 3 − 3 x y + 6 x Differentiate f ( x , y ) with respect to x , we get f x ( x , y ) = d d x ( y 3 − 3 x y + 6 x ) f x ( x , y ) = 0 − 3 y + 6 f x ( x , y ) = 6 − 3 y Differentiate f ( x , y ) with respect to y , we get f y ( x , y ) = d d y ( y 3 − 3 x y + 6 x ) f y ( x , y ) = 3 y 2 − 3 x + 0 f y ( x , y ) = 3 y 2 − 3 x To calculate the critical point, put f x ( x 0 , y 0 ) = 0 and f y ( x 0 , y 0 ) = 0 , we have 6 − 3 y = 0 − 3 y = − 6 y = 2 and 3 y 2 − 3 x = 0 y 2 − x = 0 x = y 2 Thus, critical point is ( 4 , 2 )

6th Edition

ISBN: 9781119399322

Author: Hughes-Hallett

Publisher: WILEY

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Chapter 8.5, Problem 18P

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To find all critical points for below function and determine each point as local maxima, local minima and saddle point.

f(x,y)=y3−3xy+6x

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Chapter 8.5, Problem 17P

Chapter 8.5, Problem 19P

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To find all critical points for below function and determine each point as local maxima , local minima and saddle point. f ( x , y ) = y 3 − 3 x y + 6 x

Solution Summary: The author explains how to find all critical points for a function and determine each point as local maxima, local minima and saddle point.

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To find all critical points for below function and determine each point as local maxima, local minima and saddle point.

f(x,y)=y3−3xy+6x

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Chapter 8 Solutions

To find all critical points for below function and determine each point as local maxima , local minima and saddle point. f ( x , y ) = y 3 − 3 x y + 6 x

Solution Summary: The author explains how to find all critical points for a function and determine each point as local maxima, local minima and saddle point.

Videos

To find all critical points for below function and determine each point as local maxima, local minima and saddle point. f(x,y)=y3−3xy+6x

Want to see the full answer?

Chapter 8 Solutions

To find all critical points for below function and determine each point as local maxima, local minima and saddle point.

f(x,y)=y3−3xy+6x