Explanation Given: The function is f ( x ) = x 8 3 − 4 x 5 3 . Calculation: Calculate f ′ ( x ) as follows. f ′ ( x ) = d d x ( x 8 3 − 4 x 5 3 ) = 8 3 ( x 5 3 ) − 4 ( 5 3 ) x 2 3 = 8 3 x 5 3 − 20 3 x 2 3 Evaluate f ″ ( x ) from f ′ ( x ) . f ″ ( x ) = d d x ( 8 3 x 5 3 − 20 3 x 2 3 ) = 8 3 ( 5 3 ) x 2 3 − 20 3 ( 2 3 ) x − 1 3 = 40 9 x 2 3 − 40 9 ( 1 x 1 3 ) = 40 ( x − 1 ) 9 x 1 3 Thus, the value of f ″ ( x ) is 40 ( x − 1 ) 9 x 1 3 . Equate f ″ ( x ) to 0 to find the critical points. 40 9 ( x − 1 ) x − 1 3 = 0 x − 1 3 = 0 or x − 1 = 0 x = 0 or x = 1 Thus, the critical points are 0 and 1. Divide the domain of the function into the following intervals. ( − ∞ , 0 ) , ( 0 , 1 ) and ( 1 , ∞ ) . Consider a test point in each interval to determine the concavity of the curve. Let x = − 1 be a test point in the interval ( − ∞ , 0 ) . Substitute the value x = − 1 in f ″ ( x ) . f ″ ( − 1 ) = 40 ( − 1 − 1 ) 9 ( − 1 ) 1 3 = − 80 − 9 = 80 9 > 0 Therefore, the curve is concave upward on the interval ( − ∞ , 0 )

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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The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 13.3, Problem 45E

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To find: The inflection points and to determine where the given curve is concave upward or downward.

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Chapter 13.3, Problem 44E

Chapter 13.3, Problem 46E

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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 inflection points and to determine where the given curve is concave upward or downward.

Solution Summary: The author explains that the curve is concave upward or downward on the intervals (0,1).

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To find: The inflection points and to determine where the given curve is concave upward or downward.

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Chapter 13 Solutions