Explanation Given: Function f ( x , y ) = x 6 + y 6 . 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 ( 0 , 0 ) f y y ( 0 , 0 ) − f x y ( 0 , 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 ) = x 6 + y 6 Differentiate f ( x , y ) with respect to x , we get f y ( x , y ) = d d y ( x 6 + y 6 ) f y ( x , y ) = 6 y 5 + 0 f y ( x , y ) = 6 y 5 6 y 5 = 0 y = 0 Differentiate f ( x , y ) with respect to y , we get f y ( x , y ) = d d y ( x 6 + y 6 ) f y ( x , y ) = 6 y 5 + 0 f y ( x , y ) = 6 y 5 To calculate the critical point, put f x ( x 0 , y 0 ) = 0 and f y ( x 0 , y 0 ) = 0 , we have 6 x 5 = 0 x = 0 and 6 y 5 = 0 y = 0 Thus, critical point is ( 0 , 0 )

Applied Calculus, 6e WileyPLUS + Loose-leaf

6th Edition

ISBN: 9781119408901

Author: Deborah Hughes-Hallett, Andrew M. Gleason, Patti Frazer Lock, Daniel E. Flath

Publisher: Wiley (WileyPLUS Products)

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Chapter 8.5, Problem 10P

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

f(x,y)=x6+y6

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Chapter 8.5, Problem 9P

Chapter 8.5, Problem 11P

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

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

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

f(x,y)=x6+y6

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Chapter 8 Solutions

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

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

Concept explainers

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To find all critical points for below function and determine each point aslocal maxima, local minima and saddle point. f(x,y)=x6+y6

Want to see the full answer?

Chapter 8 Solutions

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

f(x,y)=x6+y6