Extreme Value Theorem (EVT) - Expii

The extreme value theorem (EVT) says that if f is continuous on a closed interval [a,b], then there exists an input x* that maximizes f: namely, f(x*) ≥ f(x)$ for all x in [a,b]. Similarly, there exists an input that minimizes f. (Neither the maximizer or the minimizer needs to be unique -- do you see why?) This is not necessarily true if f is only continuous on an open (or half-open) interval: for instance, f could be unbounded; but even for f(x) = x (which is bounded) on the open interval (0,1), you can see that neither a maximizer nor a minimizer exists. Anyways, the EVT will come up later when we talk about Rolle's theorem. EVT also provides the theoretical basis for optimization, an important real-life concept in economics, decision-making, and elsewhere.