I don't know what you mean by "Minimum/Maximum Radius". The minimum or maximum radius to do what? The formula you post is for finding the area of a circle segment of any radius. There is no "minimum or maximum radius" except for the obvious minimum: 0 radius with 0 area. But the radius can be as large as you please making the area as large as you please.
As for the math problem website you give, Are you looking for the angle for the opening that will make that circular gutter as large as possible for a given Circumference? I think you may be overlooking the "for a given circumference" condition. Without that we can make the area as large as we wish by taking the angle closer and closer to 0.
Start with a circle of radius R. It has circumference and area . If we cut an angle , measured in radians, out of the circle, we cut out of the circumference, leaving and we cut, using that formula you found, leaving a cross-section area of .
For a given circumference, , we have so that .
Since sine is periodic with period , [tex]sin(\theta)= sin(2\pi- C/R)= sin(-C/R)= -sin(C/R) so the area is given by .
Now, it is not a matter of differentiating that with respect to to determine the "optimum" . As I said, that would be to maximize the cross section area. The problem is simply to compare that, for a given circumference C with the areas the same cross section lengths of the other two figures to see which gives the greatest area. There is no calculus required.