Expii

# 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".