I've got this rather simple problem; I get the right solution, the trouble is that I reach it by intuition rather than understanding why I should do as I do.

I should also point out that I know how to do this by simply taking the derivative of the function, but right now I'm rehearsing a chapter that haven't reached that point yet and I want to actually understand this too.

So, to boil it down I want to find the integer x > 0 for which

reach an absolute minimum. What I do is this:

And here simply observe that for x > 0 T has an absolute minimum at 25, which it reaches when the squared expression is zero, i.e. x = 5, which is the correct answer according to the key.

Now I only want to understand why this works. A little help here would be greatly appreciated!