Explanation Given: The function is f ( x ) = ln x + ln ( 4 − x 2 ) , ( 0 , 2 ) . Formula used: To find the critical points substitute f ′ ( x ) = 0 . Second derivative test: If f ′ ( c ) = 0 and f ″ ( c ) exists, then f ( c ) is a local maximum if f ″ ( c ) < 0 . f ( c ) is a local minimum if f ″ ( c ) > 0 . The test fails if f ″ ( c ) = 0 . When test fails then use the first derivative test. Calculation: The function is f ( x ) = ln x + ln ( 4 − x 2 ) , ( 0 , 2 ) . We have to find the critical points and apply the second derivative test. First, find the derivative. f ′ ( x ) = 1 x + 1 4 − x 2 d d x ( 4 − x 2 ) = 1 x + − 2 x 4 − x 2 = 1 x − 2 x 4 − x 2 Now, find the second derivative. f ″ ( x ) = − 1 x 2 − ( 4 − x 2 ) d d x 2 x − 2 x d d x ( 4 − x 2 ) ( 4 − x 2 ) 2 = − 1 x 2 − 2 ( 4 − x 2 ) − 2 x ( − 2 x ) ( 4 − x 2 ) 2 = − 1 x 2 − 8 − 2 x 2 + 4 x 2 ( 4 − x 2 ) 2 = − 1 x 2 − 8 + 2 x 2 ( 4 − x 2 ) 2 Now, find the critical points by substitute f ′ ( x ) = 0 ...

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

3rd Edition

ISBN: 9781319048518

Author: Colin Adams Jon Rogawski

Publisher: Macmillan Higher Education

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Question

Chapter 4.4, Problem 40E

To determine

Find the critical points. Also, apply the second derivative test (or state that it fails).

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Chapter 4.4, Problem 39E

Chapter 4.4, Problem 41E

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Chapter 4 Solutions

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Prob. 12E Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Prob. 15E Ch. 4.1 - Prob. 16E Ch. 4.1 - Prob. 17E Ch. 4.1 - Prob. 18E Ch. 4.1 - Prob. 19E Ch. 4.1 - Prob. 20E Ch. 4.1 - Prob. 21E Ch. 4.1 - Prob. 22E Ch. 4.1 - Prob. 23E Ch. 4.1 - Prob. 24E Ch. 4.1 - Prob. 25E Ch. 4.1 - Prob. 26E Ch. 4.1 - Prob. 27E Ch. 4.1 - Prob. 28E Ch. 4.1 - Prob. 29E Ch. 4.1 - Prob. 30E Ch. 4.1 - Prob. 31E Ch. 4.1 - Prob. 32E Ch. 4.1 - Prob. 33E Ch. 4.1 - Prob. 34E Ch. 4.1 - Prob. 35E Ch. 4.1 - Prob. 36E Ch. 4.1 - Prob. 37E Ch. 4.1 - Prob. 38E Ch. 4.1 - Prob. 39E Ch. 4.1 - Prob. 40E Ch. 4.1 - Prob. 41E Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.1 - Prob. 46E Ch. 4.1 - Prob. 47E Ch. 4.1 - Prob. 48E Ch. 4.1 - Prob. 49E Ch. 4.1 - Prob. 50E Ch. 4.1 - Prob. 51E Ch. 4.1 - Prob. 52E Ch. 4.1 - Prob. 53E Ch. 4.1 - Prob. 54E Ch. 4.1 - Prob. 55E Ch. 4.1 - Prob. 56E Ch. 4.1 - 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