Explanation Given: The polynomial has its relative maxima as ( 2 , 4 ) , relative minima as ( 4 , 2 ) and its inflection point as ( 3 , 3 ) . Formula used: For a function f that is twice differentiable on an open interval I , If f ' ( c ) = 0 for some c , and, If f ' ' ( c ) > 0 the function f has relative minima at c If f ' ' ( c ) < 0 the function f has relative maxima at c . If f ' ' ( c ) = 0 , the test fails. Calculation: First differentiate the provided polynomial a x 3 + b x 2 + c x + d p ' ( x ) = 3 a x 2 + 2 b x + c The minima and maxima of the polynomial lie at points 4 and 2 respectively. At both these points, the first derivative would disappear. This would imply: 3 a ( 4 ) 2 + 2 b ( 4 ) + c = 0 48 a + 8 b + c = 0 And, 3 a ( 2 ) 2 + 2 b ( 2 ) + c = 0 12 a + 4 b + c = 0 Now differentiate the polynomial again: p ' ' ( x ) = 6 a x + 2 b The inflection point occurs when x takes a value 3. This would imply that the second derivative would disappear at 3. This gives: 6 a ( 3 ) + 2 b = 0 18 a + 2 b = 0 b = − 9 a This can be used to convert the previously written two equations from three variables into equation in two variables

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 4.4, Problem 78E

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To calculate: The values of a, b, c and d for the polynomial ax3+bx2+cx+d that satisfies the provided conditions, The polynomial has its relative maxima as (2,4), relative minima as (4,2) and its inflection point as (3,3).

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Chapter 4.4, Problem 77E

Chapter 4.4, Problem 79E

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Chapter 4 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 4.1 - The Value of the Derivative at Relative Extrema In...Ch. 4.1 - Prob. 2E Ch. 4.1 - The Value of the Derivative at Relative Extrema In...Ch. 4.1 - Prob. 4E Ch. 4.1 - The Value of the Derivative at Relative Extrema In...Ch. 4.1 - The Value of the Derivative at Relative Extrema In...Ch. 4.1 - Approximating Critical Numbers In Exercises 11-14,...Ch. 4.1 - Prob. 8E Ch. 4.1 - Approximating Critical Numbers In Exercises 11-14,...Ch. 4.1 - Approximating Critical Numbers In Exercises 11-14,...

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The values of a , b , c and d for the polynomial a x 3 + b x 2 + c x + d that satisfies the provided conditions, The polynomial has its relative maxima as ( 2 , 4 ) , relative minima as ( 4 , 2 ) and its inflection point as ( 3 , 3 ) .

Solution Summary: The author explains that the polynomial has its relative maxima as (2,4).

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To calculate: The values of a, b, c and d for the polynomial ax3+bx2+cx+d that satisfies the provided conditions, The polynomial has its relative maxima as (2,4), relative minima as (4,2) and its inflection point as (3,3).

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Chapter 4 Solutions