A rectangle is inscribed with its base on the x -axis and its upper corners on the parabola y=2−x^2 . What are the dimensions of such a rectangle with the greatest possible area

Answer 1

We have that y(x) = 2 - x^2

We note that area (A) is given by

A = 2x*y(x) = 2x * (2 - x^2) = 4x - 2*x^3

Taking the derivative of A with respect to x and setting it equal to 0.0 to get the maximum:

dA/dx = 4 - 6±*x^2 = 0

Solving for x:

x = ±sqrt(2/3)

So the maximum area is

Amax = 2 * sqrt(2/3) * (2 - sqrt(2/3)^2) = 2.177.

Note that using the -sqrt(2/3) solution gives a negative area.