while working on some computations, I figured out that my arguments supply a proof for
the arithmetic and geometric mean inequality by using calculus. I wanted to share here.
Assertion. For all show that with equality if and only if .
Proof. The case is trivial.
Let and define
We shall prove that for all , with equality if and only if .
In this case, we may find such that (in particular, ), which yields
, where for .
As , we have to show that for all , with equality if and only if .
We can compute that
and for all .
The function is therefore convex and this shows that is the global minimum value
since . Finally, for all , with equality if and only if .