This is just the extreme value theorem. Since f is continuous and positive on a compact set [a,b], it achieves a positive minimum in [a,b].
An example of a function for which it fails is on or on .
For example, there is a theorem that says that for a continuous function on a compact interval [a;b], there are , such that , for all .
So, if you can use that theorem, you can just put , but if you cannot use that theorem, you have to (essentially) prove it.