hHow to use maximum principle of $\frac{1}{f}$ in points where $f(x) = 0$

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Suppose That I am trying to use the maximum principle on $\frac{1}{f(x)}$ in order to use the minimum principle over $f(x)$. How do I deal with points in which $f(x) = 0$?

I didn't encounter a question like that, I am just wondering.

Thanks!