Question

Find the rectangle of largest area that can be inscribed inside the region bounded by y = 1 − x^2 and the x-axis, with one side on the x-axis.

Answer 1

Let the point of contact in quad I be (x,y), then the point of contact
in quad III is (-x,y)
the length of the rectangle is 2x, and its height is y
Area = 2xy
= 2x(1-x^2)
= 2x - 2x^3
dArea/dx = 2 - 6x^2
= 0 for a max of area
6x^2 = 2
x^2 = 1/3
x = ± 1/√3 , then y = 1 - (1/√3)^2 = 1 - 1/3 = 2/3

largest area = 2xy
= 2(1/√3)(2/3)
= 4/(3√3) = 4/(3√3) * √3/√3 = 4√3 / 9