Explanation Given Information: The provided function is, y = x 3 − 3 x 2 + 6 Formula used: To find relative maxima and minima of a function, 1. Find the critical values of a function. 2. Substitute the critical values into f ( x ) to find the critical points. 3. Evaluate f ″ ( x ) at each critical value for which f ′ ( x ) = 0 . (a) If f ″ ( x 0 ) < 0 , a relative maximum occurs at x 0 . (b) If f ″ ( x 0 ) > 0 , a relative minimum occurs at x 0 . (c) If f ″ ( x 0 ) = 0 or f ″ ( x 0 ) is undefined, the second derivative test fails and then use the first derivative test. Calculation: Consider the provided function, y = x 3 − 3 x 2 + 6 Now, obtain the first derivative as, y = x 3 − 3 x 2 + 6 y ′ = 3 x 2 − 6 x = 3 x ( x − 2 ) Now, to obtain the critical values, set y ′ = 0 as, 3 x ( x − 2 ) = 0 x = 2 And, x = 0 Now, substitute 0 for x in the provided equation as, y = 0 − 0 + 6 = 6 And, substitute 2 for x in the provided equation as, y = ( 2 ) 3 − 3 ( 2 ) 2 + 6 = 8 − 12 + 6 = 2 Thus, the critical points are ( 0 , 6 ) and ( 2 , 2 ) . Now, obtain the second derivative as, y ′ = 3 x 2 − 6 x Differentiate the function with respect to x, y ″ = 6 x − 6 To obtain point of inflection, set y ″ = 0 as, 6 x − 6 = 0 6 x = 6 x = 1 Now, substitute 1 for x in the provided equation as, y = 1 − 3 + 6 = − 2 + 6 = 4 Therefore, the point of inflection is ( 1 , 4 )

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 10.2, Problem 16E

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Chapter 10.2, Problem 15E

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Chapter 10 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 10.1 - 1. The x- values of critical points are found...Ch. 10.1 - Prob. 2CP Ch. 10.1 - Prob. 3CP Ch. 10.1 - Prob. 4CP Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - In Problems 1 and 2, use the indicated points on...Ch. 10.1 - 3. Use the graph of in Problem 1 to identify at...Ch. 10.1 - 4. Use the graph of in Problem 2 to identify at...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...Ch. 10.1 - In Problems 5 and 6, use the sign diagram for to...

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The relative minimum, relative maximum and points of inflection for the provided function y = x 3 − 3 x 2 + 6 .

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