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.