The statement is true if is constant. So suppose that is not constant. Then there exists some
,
such that
.
Let's say that has a larger value. Due to
fact,
there exists some
,
where the function attains its
maximum.
This point is not on the border. For this , we have
,
due to
fact.
![{\displaystyle {}x\in {]a,b[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d5258fad15c7c5b7180a8778c8e887d86864ef)
![{\displaystyle {}c\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e96620a8d7941d7b882a2148f8703b82848ecad)
