Explanation Given: The following function is given: f ( x , y ) = x − y 2 − ln ( x + y ) Let P = ( a , b ) be a critical point of f ( x , y ) . Let the second order derivatives f x x , f y y , f x y are continuous near P . Let D be the discriminant of the function f ( x , y ) and is defined as D ( x , y ) = f x x f y y − f x y 2 (a) If D > 0 , then f ( a , b ) is a local minimum or local maximum. (b) If D < 0 , then ( a , b ) is a saddle point. (c) If D = 0 , then the test is inconclusive. (d) If D > 0 and f x x ( a , b ) > 0 , then f ( a , b ) is a local minimum. (e) If D > 0 and f x x ( a , b ) < 0 , then f ( a , b ) is a local maximum. Calculation: Consider the following function: f ( x , y ) = x − y 2 − ln ( x + y ) Differentiating, f x = 1 − 1 x + y f y = − 2 y − 1 x + y For critical points, putting f x = 0 , f y = 0 1 − 1 x + y = 0 and - 2 y - 1 x + y = 0 x + y = 1 , 2 y = − 1

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

See similar textbooks

Question

Chapter 14.7, Problem 21E

To determine

The critical points of the function.

Whether they are local minimum, local maximum or saddle points using the Second Derivative Test.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.7, Problem 20E

Chapter 14.7, Problem 22E

Students have asked these similar questions

nd ave a ction and ave an 48. The domain of f y=f'(x) x 1 2 (= x<0 x<0 = f(x) possible. Group Activity In Exercises 49 and 50, do the following. (a) Find the absolute extrema of f and where they occur. (b) Find any points of inflection. (c) Sketch a possible graph of f. 49. f is continuous on [0,3] and satisfies the following. X 0 1 2 3 f 0 2 0 -2 f' 3 0 does not exist -3 f" 0 -1 does not exist 0 ve tes where X 0 < x <1 1< x <2 2

Numerically estimate the value of limx→2+x3−83x−9, rounded correctly to one decimal place. In the provided table below, you must enter your answers rounded exactly to the correct number of decimals, based on the Numerical Conventions for MATH1044 (see lecture notes 1.3 Actions page 3). If there are more rows provided in the table than you need, enter NA for those output values in the table that should not be used. x→2+ x3−83x−9 2.1 2.01 2.001 2.0001 2.00001 2.000001

Find the general solution of the given differential equation. (1+x)dy/dx - xy = x +x2

Chapter 14 Solutions

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

Knowledge Booster

The critical points of the function. Whether they are local minimum , local maximum or saddle points using the Second Derivative Test.

The critical points of the function. Whether they are local minimum, local maximum or saddle points using the Second Derivative Test.

Want to see the full answer?

Chapter 14 Solutions

The critical points of the function.

Whether they are local minimum, local maximum or saddle points using the Second Derivative Test.