It should be noted that in reality you can not derive anything meaningful at all from the provided information. While it's true that mathematically speaking, $f$ must have a local maximum at $t=10$, the information is pretty useless because you cannot distinguish between a small bump and a major peak.
The values of $f$ at $t=0,1,2,\ldots$ hours could be
i.e. the local maximum at $t=10$ could be a small additional burp. Or, of course, the values of $f$ could be
i.e. the flow rate rises exponentially with a small irregularity around $t=10$. If it's your house that is potentially in the way of this lava flow, then believe me, there's a huge difference between those two cases...
Plus, there's no information about the time resolution of $f$. Since $f$ is a flow rate, it's basically meaningless without that. Say you wanted to know how much lava has errupted in total 5 hours after the erruption started? Could you simply compute $f(0)+\ldots+f(5)$ to get an approximate result? You can't! For that to work, you'd need to know that $f(t)$ is the average flow over one-hour intervals. As it stands, $f$ could be the average flow-rate over one-minute intervals. In which case, of course, $f(0)+\ldots+f(5)$ is probably more than a magnitude from the actual total errupted mass between $t=0$ and $t=5$.
In conclusion, this is an example of an absolutely horrible piece of mathematical exercise. It takes mathematicaly concepts, extremal points and derivatives, and applies them to a practical problem with total disregard of whether they are applicable or not.
Mathematics has tons of tools to deal with partial information about functions, measurement errors and extremal point estimation, which allow you to derive actual and dependable results. Trying to find extremal points from sampled values by methods developed for differentiable function is not one of them!