Let and assume is a function with continuous in such that

Furthermore, assume is maximal at a point (i.e., does not have its maximum at either endpoint of the interval). Prove that

*Proof.* Since attains its maximum on the interval we know there is some such that . Then,

Evaluating these integrals separately, we have (by the first fundamental theorem of calculus, which is permissible since is continuous by hypothesis)

Now, we use the bound for all ,

Next, we evaluate the second integral,

And so,

Therefore,