Explanation Given: The function is f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 . Approach: Need to identify all relevant features of the graph of this function, i.e., its domain, critical points, regions over which it is increasing or decreasing, points of inflection, regions over which it is concave up or concave down, intercepts, and asymptotes. After that graph the function, showing all these features. Calculation: Consider the function. f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 ……. (1) Differentiate Equation (1) with respect to x . f ′ ( x ) = − 6 x 2 − 18 x + 108 = − 6 ( x 2 + 3 x − 18 ) = − 6 ( x + 6 ) ( x − 3 ) …… (2) Substitute 0 for f ′ ( x ) in Equation (2) 0 = − 6 ( x + 6 ) ( x − 3 ) x = − 6 , 3 Substitute − 6 for x in Equation (1). f ( − 6 ) = − 2 ( − 6 ) 3 − 9 ( − 6 ) 2 + 108 ( − 6 ) − 10 = − 550 Substitute 3 for x in Equation (1). f ( 3 ) = − 2 ( 3 ) 3 − 9 ( 3 ) 2 + 108 ( 3 ) − 10 = 179 Therefore, the critical points for the function are ( − 6 , − 550 ) and ( 3 , 179 )

2nd Edition

ISBN: 9780321964038

Author: GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher: Pearson Addison Wesley,

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Chapter 5.4, Problem 3E

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The function by considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection point, regions where the function is concave upward or concave downward, intercepts where possible and asymptotes where applicable.

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Chapter 5.4, Problem 2E

Chapter 5.4, Problem 4E

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To Graph: The function by considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection point, regions where the function is concave upward or concave downward, intercepts where possible and asymptotes where applicable.

Solution Summary: The author analyzes the graph of the function by considering its domain, critical points, symmetry, inflection point, regions where it is increasing or decreasing, intercepts, and asymptotes.