A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 10−x2. What are the dimensions of such a rectangle with the greatest possible area?
We first formulate everything in mathematical terms: if is the distance from the origin of one lower corner of the rectangle, then the base of the rectangle is , and the height of the rectangle is . Its area is therefore
To maximize , we remember that is differentiable everywhere and approaches at the boundary points of its domain, and therefore that will attain an extremum at some point at which .