A rectangular field is to be fenced on all four sides with exactly 40 meters of fencing material.
Let x meters denote the length of one side of the rectangle, and let A(x)=x(20−x) represent the area (in square meters) enclosed by the rectangle in terms of x.
(a) Describe the practical meaning of A′(x) for this situation, including its units.
(b) Explain why any interior maximum or minimum value of the area must occur at a value of x for which A′(x)=0.
(c) Show that A′′(x)<0 at the critical point of A(x) and interpret this fact in the context of the field.