   Chapter 3.3, Problem 63E

Chapter
Section
Textbook Problem

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

To determine

To show:

The inflection points of the curve y=xsinx lie on the curve y2x2+4=4x2

Explanation

1) Concept:

i) Inflection point is the point on the curve where the curve changes from concave upward to concave downward or from concave downward to concave upward that is, f"(x)=0

ii) The inflection point lies on the given curve y=xsinx, using concept i) form a curve y2(4+x2)=4x2 shows inflection point lie on it.

2) Calculation:

y=xsinx

Differentiate y,

y'=ddxxsinx=xcosx+sinx

Again differentiate y',

y"=ddx(xcosx+sinx)

=xddxcosx+cosxddxx+ddxsinx

=-xsinx+cosx+cosx

=-xsinx+2cosx

The inflection point lies on the curve y=xsinx therefore, y"=0

-xsinx+2cosx=0

2cosx=xsinx

2cosx=y (1)

Squaring on both sides,

2cosx2=xsinx2

4

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