# For a rectangle having its base on x- axis and two vertices on the curve y = e − x 2 , the area of the rectangle is maximum when the two vertices are the inflection point of the curve y . ### 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 1P
To determine

## To show: For a rectangle having its base on x-axis and two vertices on the curve y=e−x2 , the area of the rectangle is maximum when the two vertices are the inflection point of the curve y.

Expert Solution

### Explanation of Solution

Proof:

Given:

A rectangle has its base on x-axis and two vertices on the curve y=ex2 .

Area of the rectangle under the curve from xtox is given by, A=2xex2 where x>0 .

First find the values of x for which the area is maximum. If the same points are inflection points of the curve then the hypothesis will hold.

Find the point at which the function A=2xex2 is maximum.

Differentiate A with respect to x,

A=2(2x2)ex2+2ex2=2ex24x2ex2=2ex2(12x2)

Set A=0 to get,

2ex2(12x2)=012x2=0x=±12

Hence, the value of the function A=2xex2 is maximum for x=±12 . Thus, the vertices of the rectangle are x=±12 .

Find the inflection points of the curve y=ex2 .

Differentiate with respect to x.

y=2xex2

Again differentiate with respect to x.

y=2ex2(2x21)

Equate y to zero which gives inflection points of the curve.

y=02ex2(2x21)=02x21=0x=±12

Therefore, the inflection points are x=±12 .

Thus, it is proved that the inflections points are at the vertices.

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