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: 9781119399322

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.

Slope

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 4.2, Problem 39P

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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.

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Chapter 4.2, Problem 38P

Chapter 4.2, Problem 40P

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Chapter 4 Solutions

APPLIED CALCULUS (WILEY PLUS)

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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

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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

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.