Explanation Definition used: Increasing function: Let f be a function defined on some interval with derivatives at each point. Then the function is said to be increasing if f ′ ( x ) > 0 holds for each x in the interval. Constant function: Let f be a function defined on some interval with derivatives at each point. Then the function is said to be constant if f ′ ( x ) = 0 holds for each x in the interval. Decreasing function: Let f be a function defined on some interval with derivatives at each point. Then the function is said to be decreasing if f ′ ( x ) < 0 holds for each x in the interval. 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 ) . Calculation: The given function is f ( x ) = − 2 x x + 2 . Compute the derivative of the function f ( x ) = − 2 x x + 2 . f ′ ( x ) = ( x + 2 ) ⋅ ( − 2 ) − ( − 2 x ) ⋅ ( 1 ) ( x + 2 ) 2 = − 2 ( x + 2 ) + 2 x ( x + 2 ) 2 = − 2 x − 4 + 2 x ( x + 2 ) 2 = − 4 ( x + 2 ) 2 . Set f ′ ( c ) = 0 to find the critical number c . − 4 ( c + 2 ) 2 = 0 Observe that f ′ ( x ) is never 0 and fails to exist at x = − 2 . Therefore, there are no critical points since, f is undefined for x = − 2 . Consider the interval notation. ( − ∞ , − 2 ) ∪ ( − 2 , ∞ ) Consider a point from each interval and find the corresponding value of the function. Consider a point x = − 3 on the interval ( − ∞ , − 2 ) and find the value of the function as follows. f ′ ( − 3 ) = − 4 ( ( − 3 ) + 2 ) 2 = − 4 ( − 1 ) 2 = − 4 < 0 Consider a point x = − 1 on the interval ( − 2 , ∞ ) and find the value 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 13.1, Problem 4YT

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To find: The intervals in which the function is increasing or decreasing and graph the function.

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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 intervals in which the function is increasing or decreasing and graph the function.

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Chapter 13 Solutions