Graph of the function y = f(t), t being the rotation angle of a rectangle about its gravity center and y being the area of the enclosing frame E_{1}...E_{4}. y obtains its maximum value when the enclosing is a square. The minimum of y occurs at multiples of π/2. How to explain that at the maximums we have tangents whereas at minimums we have only left/right tangents?

C controls the rotation angle. Switch to the "Select on contour" tool (press CTRL+2), catch and move C, to see the variation of the area (point X on the graph of function f(t)).