Explanation Given: Function f ( x , y ) = x 2 + y 2 + 6 x − 10 y + 8 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 ) = x 2 + y 2 + 6 x − 10 y + 8 Differentiate f ( x , y ) with respect to x , we get f x ( x , y ) = d d x ( x 2 + y 2 + 6 x − 10 y + 8 ) f x ( x , y ) = 2 x + 0 + 6 − 0 + 0 f x ( x , y ) = 2 x + 6 Differentiate f ( x , y ) with respect to y , we get f y ( x , y ) = d d y ( x 2 + y 2 + 6 x − 10 y + 8 ) f y ( x , y ) = 0 + 2 y + 0 − 10 + 0 f y ( x , y ) = 2 y − 10 To calculate the critical point, put f x ( x 0 , y 0 ) = 0 and f y ( x 0 , y 0 ) = 0 , we have 2 x + 6 = 0 2 x = − 6 x = − 3 and 2 y − 10 = 0 2 y = 10 y = 5 Thus, critical point is ( − 3 , 5 ) Calculating whether the critical point is local minima, local maxima or saddle point

6th Edition

ISBN: 9781119399353

Author: DA

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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Chapter 8.5, Problem 17P

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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)=x2+y2+6x−10y+8

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Chapter 8.5, Problem 16P

Chapter 8.5, Problem 18P

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A tank initially contains 50 gal of pure water. Brine containing 3 lb of salt per gallon enters the tank at 2 gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min. Thus, the tank is empty after exactly 50 min. (a) Find the amount of salt in the tank after t minutes. (b) What is the maximum amount of salt ever in the tank?

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By using the numbers -5;-3,-0,1;6 and 8 once, find 30

Chapter 8 Solutions

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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 ) = x 2 + y 2 + 6 x − 10 y + 8

Solution Summary: The author explains the formula used to find all critical points for a function and determine each 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)=x2+y2+6x−10y+8

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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 ) = x 2 + y 2 + 6 x − 10 y + 8

Solution Summary: The author explains the formula used to find all critical points for a function and determine each 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)=x2+y2+6x−10y+8

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)=x2+y2+6x−10y+8