Find the dimensions of the shaded region so that its area is maximized.
x=
y=

1.  Find the equation of the line AB.  For reference, the answer is y=(-2/3)x+2.
2.  Derive a formula for the area of the shaded rectange.  It is A=xy (where x is the length and y is the height).
3.  Replace "y" in A=xy with the formula for y:  y= (-2/3)x+2:
       
                 A=x[(-2/3)x+2]    This is a formula for Area A in terms of x only.
4.  Since we want to maximize the shaded area, we take the derivative with respect to x of    A=x[(-2/3)x+2] , or, equivalently, A=(-2/3)x^2 + 2x.
This results in   (dA/dx) = (-4/3)x + 2.
5.  Set this result = to 0 and solve for the critical value:  

(dA/dx) = (-4/3)x + 2=0, or (4/3)x=2   This results in x=(3/4)(2)=3/2

6.  Verify that this critical value x=3/2 does indeed maximize the area function.
7.  Determine the area of the shaded rectangle for x=3/2, using the previously-derived formula          A=(-2/3)x^2 + 2x.

The result is the max. area of the shaded rectangle.