What is the area of the largest rectangle?

Question

Find the area of the largest rectangle having two vertices on the 𝑥 − 𝑎𝑥𝑖𝑠 and two vertices above the 𝑥 − 𝑎𝑥𝑖𝑠 and on the parabola with equation 𝑦 = 9 − 𝑥²

Answer 1

it's a downward opening parabola with vertex = maximum point = (0,9)

half the area of a rectangle would be xy = x(9-x^2) = 9x-x^3

A' = 9-3x^2 = 0

3x^2 = 9

x^2 =3

x= sqr3

y=9sqr3 -3sqr = 6sqr3

xy= sqr3(6sqr3) = 18

double it; the largest area rectangle has area = 36

It's 2sqr3 by 6sqr3

Answer 2

Draw a figure!

The area involved is 2x(9-x²).

The derivative is 18-6x² which is 0 when x=√3.

You can figure the area from that answer.