A rectangle with its base on the x-axis is to be inscribed under the graph of y=2-x^2. Find the height of the rectangle if the area is the largest possible area.

Answer 1

let the point of contact of the rectangle with the parabola in the first quadrant be P(x,y)
So the base of the rectangle is 2x and its height is y
Area = 2xy
= 2x(2-x^2)
= 4x = 2x^3
d(Area)/dx = 4 - 6x^2 = 0 for a max/min of area

6x^2 = 4
x = √(2/3)

then the height for a max area
= 2 - 2/3
= 4/3