#### To determine

**To verify:**

The value of the largest area bounded by the *x*-axis and two vertices on the curve y=e−x2.

#### Answer

Verified.

#### Explanation

**Given:**

y=e−x2

**Formulae used:**

The maximise with second derivative test f″(x)<0

The area of the bounded region is

A=Base×Height=2x×e−x2=2xe−x2

Differentiate the area with respect to *x* to get the condition of maximum area

dAdx=2xe−x2(−2x)+2e−x2=2e−x2(1−2x2)

Find the critical point by putting dAdx=0 as follows:

2e−x2(1−2x2)=0

This implies:

2e−x2=0 or 1−2x2=0

2e−x2=0 gives no solution.

So, 1−2x2=0

This implies:

2x2=1x2=12x=±12

Hence, the critical points are x=±12.

Now,

d2Adx2=−2e−x2(1−2x2)

Therefore, the value of d2Adx2 at x=±12 is:

d2Adx2=e−x2(4x3−6x)<0

Thus, the rectangle has the largest possible area when the vertices are at the points of inflection of the curve.