Let be a continous function such that
and
(1)
Prove that forall
.
Solution
Consider the function which is differentiable in
. We can easily see that
has a global minimum at
that is equal to
. Visually we have that:
Clearly as we can see it holds that forall
. Also:
(2)
Thus gives us:
Therefore . Thus
forall
The exercise can also be found in mathematica.gr