A rectangular garden will be built against a straight river, so no fencing is required along the side bordering the water. The two sides perpendicular to the river use reinforced posts that cost \12perfoot.Thesideoppositetheriverusesstandardpoststhatcost8 per foot for the first 200 feet purchased; any length beyond 200 feet of standard fencing costs \$5 per foot.
The garden must enclose 24,000 square feet of area.
(a) Let x be the length (in feet) of the sides perpendicular to the river. Express the total fence cost C(x) as a piecewise-defined function of x, indicating the domain for each piece.
(b) Compute C′(x) for every differentiable portion of C and determine all critical numbers that satisfy the domain restrictions.
(c) Identify the set of candidate values for x, evaluate C at each candidate, and determine the garden dimensions that minimize the total cost.
(d) Provide a justification that your dimensions indeed yield the minimum possible cost, making explicit reference to comparisons among all candidate values and any relevant endpoints.