To show: For a rectangle having its base on x-axis and two vertices on the curve , the area of the rectangle is maximum when the two vertices are the inflection point of the curve y.
A rectangle has its base on x-axis and two vertices on the curve .
Area of the rectangle under the curve from is given by, where .
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 is maximum.
Differentiate A with respect to x,
Set to get,
Hence, the value of the function is maximum for . Thus, the vertices of the rectangle are .
Find the inflection points of the curve .
Differentiate with respect to x.
Again differentiate with respect to x.
Equate to zero which gives inflection points of the curve.
Therefore, the inflection points are .
Thus, it is proved that the inflections points are at the vertices.
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