# The inflection points of the curve y = sin x x lie on the curve y 2 ( x 2 + 4 ) = 4 .

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 5P
To determine

## To show: The inflection points of the curve y=sinxx lie on the curve y2(x2+4)=4.

Expert Solution

### Explanation of Solution

Find the second derivative of the function y.

y=sinxxy=xcosxsinxx2y=x2(cosxxsinxcosx)(xcosxsinx)2xx4=x3sinx2x2cosx+2xsinxx4

Set y=0 to get the inflection points of the curve.

x3sinx2x2cosx+2xsinxx4=0x3sinx2x2cosx+2xsinx=02xsinx=x3sinx+2x2cosx2sinx=x2sinx+2xcosx

Simplify further as follows.

2sinx=x2sinx+2xcosx(2x2)sinx=2xcosx

Square on both sides.

(2x2)2sin2x=4x2cos2x(2x2)2sin2x=4x2(1sin2x)(44x2+x4+4x2)sin2x=4x2(4+x4)sin2xx2=4

Substituting, y=sinxx, (4+x4)y2=4.

Notice that this is nothing but the given curve.

Thus, it is proved that the inflection points of the curve y=sinxx lie on the curve y2(4+x4)=4.

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