Expert: Socrates - 11/30/2009

Question
find the area of the largest rectangle that can be inscribed in the region bounded by the x-axis, y-axis and the curve f(x)=((4)/(x+1))-1

Answer
You really need to draw a picture to understand this, but here is the solution anyway.

The vertex of the rectangle on the curve has coordinates ( x , 4/(x+1) - 1 )

The other 3 vertices are (0,0) (x,0) and ( 0 , 4/(x+1) - 1 )

The curve meets the x axis at x=3 , so x will be some number between 0 and 3

The rectangle has height 4/(x+1) - 1 and width x , so the area is (x)(4/(x+1) - 1)

We want to maximize A(x) = 4x/(x+1) - x for x in [0,3]

A'(x) = 4/(x+1)^2 - 1

The maximum occurs when the derivative is 0


4/(x+1)^2 - 1 = 0

4/(x+1)^2 = 1

4 = (x+1)^2

2 = x+1

x=1

so the area is A(1) = (4)(1)/(1+1) - 1 = 1

The maximum area is 1