   Chapter 4.3, Problem 76E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# For what values of a and b is (2, 2.5) an inflection point of the curve x2y + ax + by = 0? What additional inflection points does the curve have?

To determine

To find: The values of a and b if the curve x2y+ax+by=0 have an inflection point (2,2.5) .

Explanation

Calculation:

The given curve is, x2y+ax+by=0 .

Rewrite the given curve as,

x2y+ax+by=0y(x2+b)=ax

y=axx2+b (1)

Obtain the derivative of y.

y=ddx(axx2+b)=[(x2+b)(ax)(ax)(x2+b)(x2+b)2]=[(x2+b)a(1)(ax)(2x+0)(x2+b)2]=[a(x2+b)(ax)2x(x2+b)2]

On further simplification,

y=[ax2+ab2ax2(x2+b)2]=[abax2(x2+b)2]=ax2ab(x2+b)2

Obtain the derivative of y .

y=ddx[ax2ab(x2+b)2]=(x2+b)2(ax2ab)(ax2ab)((x2+b)2)((x2+b)2)2=(x2+b)2(2ax0)(ax2ab)2(x2+b)(2x)(x2+b)4=(x2+b)2(2ax)4x(ax2ab)(x2+b)(x2+b)4

On further simplification,

y=(x2+b)[(x2+b)(2ax)4x(ax2ab)](x2+b)4=(x2+b)(2ax)4x(ax2ab)(x2+b)3=2ax3+2xab4ax3+4xab(x2+b)3=6xab2ax3(x2+b)3

Thus, the value of y=2ax(3bx2)(x2+b)3

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 