Applied maximum and minimum problems

Question

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola . What would be the width and height of the rectangle with the greatest possible area?

Answer 1

Draw a diagram.

The upper right corner has coordinates (x, 1-x²) and the upper left corner has coordinates (-x, 1-x²). The height of the rectangle is 1-x² and the width is 2x.

Let A(x) = area of rectangle = 2x(1 - x²) = -2x³ + 2x

A'(x) = -6x² + 2 = 0 when x = 1/√3

Area is maximized when width = 2x = 2/√3 and height = 1 - x² = 2/3