Explanation Given information: a x 3 + b x 2 + c x + d ; with a critical point at x = 2 and an inflection point at ( 1 , 4 ) , and a leading coefficient of 1. Definition used: A point at which the graph of a function f changes concavity is called an inflection point. At the inflection points, f ' ' is zero or undefined. The critical points are at f ' = 0. Calculation: a x 3 + b x 2 + c x + d (Given) Let y = a x 3 + b x 2 + c x + d Differentiating with respect to ‘x’, we get y ' = 3 a x 2 + 2 b x + c y ' ' = 6 a x + 2 b Since, critical point at x = 2 , y ' ( 2 ) = 0 ⇒ 3 a ( 2 ) 2 + 2 b ( 2 ) + c = 0 ⇒ 12 a + 4 b + c = 0 Since, (1, 4) is an inflection point, we have y ' ' ( 1 ) = 0 ⇒ 6 a ( 1 ) + 2 b = 0 ⇒ 6 a + 2 b = 0 Since the leading coefficient of 1, a is the leading coefficient, a = 1

6th Edition

ISBN: 9781119275565

Author: Hughes-Hallett

Publisher: WILEY

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

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Chapter 4.2, Problem 39P

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Formula for the function ax3+bx2+cx+d with a critical point at x=2 and an inflection point at (1,4), and a leading coefficient of 1.

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Chapter 4.2, Problem 38P

Chapter 4.2, Problem 40P

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APPLIED CALCULUS-PRINT COMPANION (LL)

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Formula for the function a x 3 + b x 2 + c x + d with a critical point at x = 2 and an inflection point at ( 1 , 4 ) , and a leading coefficient of 1.

Solution Summary: The author explains the formula for the polynomial ax3+b x cx+d.

Concept explainers

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To find:

Formula for the function ax3+bx2+cx+d with a critical point at x=2 and an inflection point at (1,4), and a leading coefficient of 1.

Want to see the full answer?

Chapter 4 Solutions

Formula for the function a x 3 + b x 2 + c x + d with a critical point at x = 2 and an inflection point at ( 1 , 4 ) , and a leading coefficient of 1.

Solution Summary: The author explains the formula for the polynomial ax3+b x cx+d.

Concept explainers

Videos

To find: Formula for the function ax3+bx2+cx+d with a critical point at x=2 and an inflection point at (1,4), and a leading coefficient of 1.

Want to see the full answer?

Chapter 4 Solutions

To find:

Formula for the function ax3+bx2+cx+d with a critical point at x=2 and an inflection point at (1,4), and a leading coefficient of 1.