fence a circular land and a square land.

Question

With a wire mesh of 1000 mts divided into two parts , we want to fence a circular land and a square land.

a)Calculate the lengths of each of the parties such that the total area enclosed is maximum. b)Calculate the lengths of each of the parties such that the total area enclosed is minimal.

Here is my work:

I don´t understand why i only got a minimum some help please to solve this problem please.

Answer 1

Your calculation is fine. So the only local extremum in the interior of the valid range for $r$ is a minimum. Then where in that closed interval lies the maximum?

Answer 2

(If I had the reputation I would have simply added this as a comment to Hagen's good answer.)

Knight, reference the following link: http://mathworld.wolfram.com/FirstDerivativeTest.html

Notice the emphasis this link places on the status (open or closed) of the interval of interest.

Look closely at the images. The first case has no minimum or maximum on the open interval. But, there is a maximum and a minimum on the closed interval.

The answer to your question lies in the second case.