   Chapter 4.3, Problem 78E ### 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

# Show that the curves y = e−x and y = −e−x touch the curve y = e−x sin x at its inflection points.

To determine

To show: The curves y=ex and y=ex touch the curve y=exsinx at its inflection points.

Explanation

Proof:

The given curve is, y=exsinx .

Obtain the derivative of y.

y=(exsinx)

Apply the Product Rule,

y=ex(sinx)+sinx(ex)=excosx+sinx(ex)=ex(cosxsinx)

Obtain the derivative of y .

y(x)=ex(cosxsinx)+(cosxsinx)(ex)=ex(sinxcosx)+(cosxsinx)(ex)=ex(sinxcosxcosx+sinx)=2excosx

Set y(x)=0 and find the value of x.

2excosx=0cosx=0cosx=cos(π2+nπ)x=π2+nπ

Since ex is an increasing function, it is never zero.

Inflection points occurs where the concavity changes.

Therefore, at x=π2+nπ the curve changes its concavity.

Hence, the inflection points occurs at x=π2+nπ .

Plug x=π2 in y=exsinx ,

y(π2)=e(π2)sin(π2)=e(π2)(1)

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