Find the area of the largest rectangle

Question

Find the area of the largest rectangle with one corner at the origin, the opposite corner in the first quadrant on the graph of the parabola f(x)=1458−6x^2, and sides parallel to the axes.

The maximum possible area is=?

Answer 1

If each horizontal side of the rectangle has length x, then the vertical sides each have length f(x) = -6x² + 1458.

 

Let A(x) = area of rectangle

 

A(x) = x(-6x² + 1458) = -6x³ + 1458x

 

A'(x) = -18x² + 1458, x > 0

 

Set A'(x) equal to zero to locate the critical points of A(x):

 

    -18x²  + 1458 = 0

 

                         x² = 81   So, x = 9

 

 

Check the sign of A'(x) on both sides of the critical point:

 

    If 0 < x < 9, A'(x) > 0.  So, A(x) is increasing when 0 < x < 9.

    If x > 9, A'(x) < 0. So, A(x) is decreasing when x > 9.

 

    Therefore, A(x) has a relative maximum (and absolute                        maximum) when x = 9.

 

     Maximum area = A(9) = -6(9)³ + 1458(9) = 8748.