Explanation Theorem used: “Let the partial derivatives f x x , f y y and f x y of the function z = f ( x , y ) exists in the circular region contained in the xy -plane with center ( a , b ) and f x ( a , b ) = 0 , f y ( a , b ) = 0 , then (a) The function has relative maximum at ( a , b ) if f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 > 0 and f x x ( a , b ) < 0 . (b) The function has relative minimum at ( a , b ) if f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 > 0 and f x x ( a , b ) > 0 . (c) If f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 < 0 , then f ( a , b ) is a saddle point (neither a maximum nor a minimum). (d) If f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 = 0 , then the test fails.” Calculation: Compute the partial derivative of the function f ( x , y ) = 100 + 6 x y − 4 x 2 − 3 y 2 with respect to x . f x ( x , y ) = ∂ ∂ x ( 100 + 6 x y − 4 x 2 − 3 y 2 ) = ∂ ∂ x ( 100 ) + 6 ∂ ∂ x ( x y ) − 4 ∂ ∂ x ( x 2 ) − 3 ∂ ∂ x ( y 2 ) = 0 + 6 y − 8 x − 0 = − 8 x + 6 y That is, f x ( x , y ) = − 8 x + 6 y . Again, compute the partial derivative of the function f x ( x , y ) with respect to x . f x x ( x , y ) = ∂ ∂ x ( f x ( x , y ) ) = ∂ ∂ x ( − 8 x + 6 y ) = − ∂ ∂ x ( 8 x ) + 6 ∂ ∂ x ( y ) = − 8 That is, f x x ( x , y ) = − 8 . …… (1) Compute the partial derivative of the function f ( x , y ) with respect to y . f y ( x , y ) = ∂ ∂ y ( 100 + 6 x y − 4 x 2 − 3 y 2 ) = ∂ ∂ y ( 100 ) + ∂ ∂ y ( 6 x y ) − 4 ∂ ∂ y ( x 2 ) − 3 ∂ ∂ y ( y 2 ) = 0 + 6 x − 0 − 6 y = 6 x − 6 y That is, f y ( x , y ) = 6 x − 6 y . Again, compute the partial derivative of the function f y ( x , y ) with respect to y . f y y ( x , y ) = ∂ ∂ y ( f y ( x , y ) ) = ∂ ∂ y ( 6 x − 6 y ) = 6 ∂ ∂ y ( x ) − 6 ∂ ∂ y ( y ) = − 6 That is, f y y ( x , y ) = − 6

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 7.3, Problem 23E

To determine

To find: The local extrema of the function f(x,y)=100+6xy−4x2−3y2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 22E

Chapter 7.3, Problem 24E

Students have asked these similar questions

14 14 4. The graph shows the printing rate of Printer A. Printer B can print at a rate of 25 pages per minute. How does the printing rate for Printer B compare to the printing rate for Printer A? The printing rate for Printer B is than the rate for Printer A because the rate of 25 pages per minute is than the rate of for Printer A. pages per minute RIJOUT 40 fy Printer Rat Number of Pages 8N WA 10 30 20 Printer A 0 0 246 Time (min) X

2. y 1 Ο 2 3 4 -1 Graph of f x+ The graph gives one cycle of a periodic function f in the xy-plane. Which of the following describes the behavior of f on the interval 39 x < 41 ? (Α B The function f is decreasing. The function f is increasing. The function f is decreasing, then increasing. D The function f is increasing, then decreasing.

Depth (feet) 5- 4- 3- 2. WW www 1 D B 0 10 20 30 40 50 60 70 80 Time (hours) x A graph of the depth of water at a pier in the ocean is given, along with five labeled points A, B, C, D, and E in the xy-plane. For the time periods near these data points, a periodic relationship between depth of water, in feet, and time, in hours, can be modeled using one cycle of the periodic relationship. Based on the graph, which of the following is true? B C The time interval between points A and B gives the period. The time interval between points A and C gives the period. The time interval between points A and D gives the period. The time interval between points A and E gives the period.

Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The local extrema of the function f ( x , y ) = 100 + 6 x y − 4 x 2 − 3 y 2 .

Solution Summary: The author explains the function's local extrema and its maximum value. The function has relative minimum at (a,b).

Videos

To find: The local extrema of the function f(x,y)=100+6xy−4x2−3y2.

Want to see the full answer?

Chapter 7 Solutions