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 .
Integrating both sides from 1 to
As we have,