Bounds on functions via its derivatives

Question

Suppose, we have a function $f$ where $f$ is:

1. Contionuos.
2. Non-Negative
3. Has a derivative given by $f'$.

Can we have a bound on $f$ in terms of its derivative $f'$? That is have an inequality that says \begin{align*} f \le g(f') \end{align*} for some function $g$.

A simple example that I was able to think of is $e^x$.

Thanks. Would be gratful for any references as well.

Answer 1

$f'$ is invariant under shifts on $f$, that is $(f+k)'=f'$. Thus a bound depending on the derivative $f'$ should work for all shifts of the function $f$, which is impossible for arbitrarily large shifts $k$.

EDIT:

However, using the fundamental theorem of calculus, based around some point $x_0 \in \text{Dom }f $ we have

$$f(x)=f(x_0)+\int_{x_0}^x f'(t)dt \leq f(x_0)+M(x-x_0)$$

where $M=\max_{t \in [x_0,x]} f'(t)$. If $f'$ is bounded globally, we can use this bound for $M$. Note that this bound depends on $f'$ as well as $f(x_0)$!