Martha is fencing off a small herb garden in her backyard. if she has 20 feet of fencing and wants a rectangular garden, what length and width will give her the largest area?

First use the material to find the restrictions on the variables...

m=2x+2y

y=(m-2x)/2

Now area is:

A=xy, and using y found above we have:

A=(mx-2x^2)/2

dA/dx=(m-4x)/2

d2A/dx2=-4/2=-2

Since acceleration is a constant negative value, when velocity, dy/dx=0, it is at the point of an absolute maximum for A(x)...

dA/dx=0 only when 4x=m, x=m/4

So this indicates that x=y=m/4.

Which means that the maximum possible area is when a perfect square is formed..

In this case m/4=20/4=5.

So the garden will be a perfect 5 foot square having an area of 25 ft^2.

22.