Barrons AP Calculus - David Bock

Let (x,y) be the point in the first quadrant where the line parallel to the x-axis meets the
parabola. The area of the triangle is given by
A = xy = x(27 − x^2 ) = 27x − x^3 for 0 ≤ x ≤
Then A′ (x) = 27 − 3x^2 = 3(3 + x)(3 − x), and A′ (x) = 0 at x = 3.
Since A′ changes from positive to negative at x = 3, the area reaches its maximum there.
The maximum area is A(3) = 3(27 − 3^2 ) = 54.