A fundamental task of analysis is the search for the minimum or maximum of a function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) .Function minimumFunction maximumMinimumMaximum For simple functions, e.g., this task is straightforward, and the unique minimum is at \(x=x_0\) where the function takes the value \(f(x_0)=5\) . Theorem  7.13 , the theorem of the minimum and maximum declares that continuous functions have a minimum and maximum on every closed interval. If we compare the proof of this theorem, for example, with that of the intermediate value theorem, we notice that the intermediate value theorem is based on a constructive argument, which can be directly implemented as a numerical method. The proof of the theorem of the minimum and maximum is based on the selection of a subsequence. It is a typical result of the type there exists a point x, such that ..., but the proof does not provide us with a clear way to find this point. Indeed, the task of finding minima and maxima is much more difficult than finding roots. This is particularly true when we know very little about the function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) . Theorem  7.13 requires the continuity of f. If we do not know more about f, we will not be able to derive efficient methods.