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

Question

Answer 1

We start with the area of a rectangle A=BH=(2x)(y)=2x(1-x^2)=2x-2x^3

We must find the critical point of this function with x>0.

A'=2-6x^2=0

1-3x^2=0: 1/3=x^2: sqrt(3)/3=x

So plugging in this critical value we obtain:

A=2qrt(3)/3-2sqrt(3)/9=4sqrt(3)/9