Expii

# Optimization: Reducing to the Closed Interval Case - Expii

Finding extrema on open or unbounded intervals is similar to the process for closed intervals. For instance, say you want to optimize a function $$f$$ over the reals (which form an unbounded interval, namely $$(-\infty,\infty)$$). If, say, $$f(x) \le f(a)$$ for all $$x\le a$$ and $$f(x)\le f(b)$$ for all $$x\ge b$$ (e.g. imagine $$f$$ increasing up to $$a$$ and decreasing after $$b$$), then to find the global maxima of $$f$$ on the interval $$\mathbb{R} = (-\infty,\infty)$$, you can restrict your attention to the closed interval $$[a,b]$$, which we already know how to analyze.