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