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Is it possible to solve "minimum/maximum value of a function" problems using algebra?

First off, let me say I know these problems can be solved with derivitives in calculus, but that's not what I'm after here.

For example:

What is the minimum value of the function *f*(*x*) = *x^2 - 1?*

I know the answer is is -1, but would constructing tables such as the one below be able to reliably bring me to the right answer for problems like this one? If yes, please elaborate the method you'd use when making these tables. I found this on the internet, but can't remember how this works from my Algebra 2 class which was years ago.