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?Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution.

Calculus: Early Transcendental Functions | 6th Edition | ISBN: 9781285774770 | Authors: Ron Larson ISBN: 9781285774770 141

Solution for problem 6 Chapter 16.1

Calculus: Early Transcendental Functions | 6th Edition

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Calculus: Early Transcendental Functions | 6th Edition | ISBN: 9781285774770 | Authors: Ron Larson

Calculus: Early Transcendental Functions | 6th Edition

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Problem 6

Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution.

\(y e^{x} d x+e^{x} d y=0\)

Text Transcription:

y e^x dx+e^x dy=0

Step-by-Step Solution:

Step 1 of 5) Theorem 11—Second Derivative Test for Local Extreme Values Suppose that ƒ(x, y) and its first and second partial derivatives are continuous throughout a disk centered at (a, b) and that ƒx(a, b) = ƒy(a, b) = 0. Then i) ƒ has a local maximum at (a, b) if ƒxx 6 0 and ƒxx ƒyy - ƒxy 2 7 0 at (a, b). ii) ƒ has a local minimum at (a, b) if ƒxx 7 0 and ƒxx ƒyy - ƒxy 2 7 0 at (a, b). iii) ƒ has a saddle point at (a, b) if ƒxx ƒyy - ƒxy 2 6 0 at (a, b). iv) the test is inconclusive at (a, b) if ƒxx ƒyy - ƒxy 2 = 0 at (a, b). In this case, we must find some other way to determine the behavior of ƒ at (a, b).

Step 2 of 2

Chapter 16.1, Problem 6 is Solved
Textbook: Calculus: Early Transcendental Functions
Edition: 6
Author: Ron Larson
ISBN: 9781285774770

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?Solving an Exact Differential Equation In Exercises 5-14, determine whether the differential equation is exact. If it is,find the general solution.