I just want to ask if all shapes that have maximum area also have maximum perimeter and if all solids that have maximum volume also have maximum surface area. Why is this so? I just find it interesting that when it comes to shapes and solids, two parameters are simultaneously maximized by just forming a relationship involving two variables.
For example, the maximum area and perimeter of a rectangle with one held constant and the other held variable has the length and width equal. This relationship is the same regardless of which parameter is picked as constant and variable. What other applications of maxima and minima besides shapes and solids have this sort of event where both can be maximized or minimized using the same relationship?