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