Prove that for a fixed area, the rectangle with the minimal perimeter is a square.
Proof. Let and denote the sides of the rectangle. If the area is fixed, then is a constant, say . The perimeter of the rectangle is then a function . But, since
we can write as a function of alone,
So, to find the value of at which is minimal we take the derivative,
This has a zero at and we have
Therefore, is decreasing for and increasing for . Hence, has its minimum at . Since implies as well, we have that the perimeter is minimal when , i.e., when the rectangle is a square