Explanation Definition used: Critical point: A point whose x- coordinate is the critical number c such that either f ′ ( c ) = 0 or f ′ ( c ) does not exists and y -coordinate is f ( c ) . Increasing function: Let f be a function with some derivative defined on some interval. Then the function is said to be increasing if for any x in the interval, the inequality f ′ ( x ) > 0 holds. Decreasing function: Let f be a function with some derivative defined on some interval. Then the function is said to be decreasing if for any x in the interval, the inequality f ′ ( x ) < 0 holds. Calculation: The given function is f ( x ) = ln | x 2 − 1 | . Obtain the derivative as follows. f ′ ( x ) = ( ln | x 2 − 1 | ) ′ = 1 x 2 − 1 ⋅ 2 x = 2 x x 2 − 1 Set f ′ ( c ) = 0 and compute the value of c . 2 c c 2 − 1 = 0 Above equality is only possible when c = 0 but clearly for c = ± 1 , the derivative does not exist. Thus, the critical numbers are − 1 , 0 and 1 . These points determines the intervals, ( − ∞ , − 1 ) ∪ ( − 1 , 0 ) ∪ ( 0 , 1 ) ∪ ( 1 , ∞ ) . Consider a point from each interval and find the corresponding value of the derivative function. Consider a point x = − 2 on the interval ( − ∞ , − 1 ) and find the value derivative of the function as follows. y ′ ( − 2 ) = 2 ( − 2 ) ( − 2 ) 2 − 1 = − 4 4 − 1 = − 4 3 Consider a point x = − 1 2 on the interval ( − 1 , 0 ) and find the value derivative of the function as follows

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13, Problem 23RE

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To find: The open intervals where f(x)=ln|x2−1| increases or decreases.

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Chapter 13, Problem 22RE

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Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The open intervals where f ( x ) = ln | x 2 − 1 | increases or decreases.

Solution Summary: The author explains that the function increases or decreases on the open intervals where f(x)=mathrmln

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To find: The open intervals where f(x)=ln|x2−1| increases or decreases.

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Chapter 13 Solutions