Calculus Problem - Properties of a Function

Hello

This is my first post in this forum :)

I have a mathematical problem that I could not solve. Could you please give me some hints how to solve it?

Let be a continous and on a differentiable function with following properties:

a)

b) there exists a with for all

Now the problem is: Show that is true for all

There is a hint given but it doesn't help me :( The hint is: Consider the set for and show that the the supremum of this set is .

Thanks for help

Greetings

Re: Calculus Problem - Properties of a Function

I think I would be inclined to use the "mean value" theorem. Suppose there exist such that . Then there exist between 0 and 1 such that