The value of the largest area bounded by the x-axis and two vertices on the curve y=e−x2.
The maximise with second derivative test f″(x)<0
The area of the bounded region is
Differentiate the area with respect to x to get the condition of maximum area
Find the critical point by putting dAdx=0 as follows:
2e−x2=0 or 1−2x2=0
2e−x2=0 gives no solution.
Hence, the critical points are x=±12.
Therefore, the value of d2Adx2 at x=±12 is:
Thus, the rectangle has the largest possible area when the vertices are at the points of inflection of the curve.