Optimization: Taking Endpoints into Account - Expii

Finding extrema on open or unbounded intervals is similar to the process for closed intervals. However, when finding global extrema, you might need to be a little more careful with "boundary" analysis: what happens in the limit as you approach endpoints (for open intervals) or infinity (for unbounded intervals). For example, suppose you are trying to minimize a differentiable function \(f\) over the unbounded interval \((0,\infty)\), and you know that \(\lim_{x\to0^+} f(x) = 57\), while \(\lim_{x\to\infty} f(x) = 2017\). Then \(f\) is minimized at some critical point \(c\in (0,\infty)\) if \(f(c) < 57\), but it's possible that \(f(c) > 57\) for every critical point \(c\in (0,\infty)\), in which case \(f\) is not minimized at an actual point of \((0,\infty)\), but rather "minimized as \(x\) approaches \(0\) from the right".