**Problem:** Suppose is a real valued differentiable function defined on with . Suppose moreover satisfies

Show that for every

**Solution:** As the question doesn’t requires us to find an exact solution rather just an upper bound, we can easily find it by manipulating the given statement after establishing certain properties of .

We see that for all which means is an increasing function.

As the domain is we can say that for all .

(as )

Integrating both sides from 1 to

As we have,

Substituting

Hence Proved.