Question

What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y = 27 - x2
Did you make a sketch ?

let the vertex in quadrant I be (x,y)
then the vertex in quadratnt II is (-x,y)

the base of our rectange = 2x
and the height is y

Area = xy
= x(27 - x^2)
= -x^3 + 27x
d(area)/dx = 3x^2 - 27
= 0 for a max of area

3x^2 = 27
x^2 = 9
x = ±3
y = 27-9 = 18

largest area = 3(18) = 54 square units

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