Dear friends,

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 .