Determine smallest possible parameter set for FPT I am reading a book about complexity analysis and cannot find a way to solve a problem in that book.
The problem is, that I do not understand how to determine the smallest possible parameters, given the runtime of an algorithm. Do I just need to insert some numbers and see if the parameters exceed n at some point? Perhaps someone can explain that to me.

Consider the following exercise:


  The following O(.) expressions represent worst-case runtimes of different algorithms, where n is a measure of the input size and p, q and r are parameters (where p, q, r &lt;= n). Which of the following running times are FPT running times?
  Give the smallest possible parameter set for which the running times are FPT (if one exists) and explain.
  
  a) $O(p^q * n^2)$


Now I know that by the definition of FPT, this runtime is FPT (because we have a function of the parameters times n^constant). However, what are the smallest $p$ and $q$ in that case? 

There are some more, but I just want to understand how to solve it and then I'll try the rest myself again.

Taken from:


  Rooij, I. V., Blokpoel, M., Kwisthout, J., &amp; Wareham, T. (2019). Fixed-Parameter Tractable Time. In Cognition and Intractability: A Guide to Classical and Parameterized Complexity Analysis (p. 125). Cambridge, England: Cambridge University Press.
Recall that the runtime must be of the form $f(k) \cdot n^c$ for some constant $c$. So I believe that the exercise is asking, given that all three are at most $n$, which do you have to take as formal parameters in your analysis to achieve an FPT runtime.

So if only $p$ is a parameter, is that FPT? What if only $q$? Clearly, if both are fixed, you have FPT runtime.