Maximum area of rectangle bounded by y=x^2 and y=6-x^2

Question

Find the maximum area of a rectangle with sides parallel to the coordinate axes, that can be inscribed in the region bounded by two parabolas with equations y=x^2 and y=6-x^2

Answer 1

First, either draw the graphs or look at the two parabolas together on a graphing calculator.

 

The length of the rectangle required is 6-x² - x² = 6 - 2x² and

the width of the rectangle is 2x.

 

The area of the rectangle is 12x - 4x³.

 

The area will be maximum where the derivative 12 - 12x² = 0, i.e. where x = ±1.

That area is 8.