The paper by J. Kiefer addresses the problem of sequentially minimizing the length of an interval containing the maximum of a unimodal function on the unit interval without assuming continuity or derivatives. The solution involves a procedure that is ε-minimax among all sequential nonrandomized procedures that terminate by providing an interval containing the required point. The procedure is designed to handle observations with errors, though optimal results for this more challenging case are not yet known. The search for the maximum is described as a "second-order" search, where pairs of observations provide information. The class of functions considered, denoted as \(\mathcal{F}\), includes functions that are either strictly increasing or decreasing on specific intervals. The minimax procedure involves splitting the interval into equal parts at the point of the next observation. The paper defines a strategy \(S\) that consists of a starting point, functions for subsequent observations, and functions for selecting the final interval. The goal is to find a strategy \(S_N^*\) that minimizes the length of the final interval for any function in \(\mathcal{F}\) with \(N\) observations. The procedure \(S_N^*\) is constructed using Fibonacci numbers and specific functions to guide the observations. The proof of the minimax property is given inductively, showing that the procedure satisfies the minimax condition for \(N = 2\) and extending this to larger \(N\). The paper also discusses an alternative procedure \(S^*\) that can be more efficient in certain cases, especially when the number of observations is not fixed in advance. The references include works on stochastic estimation and stochastic approximation methods, which are relevant to the problem of finding the maximum of a function.The paper by J. Kiefer addresses the problem of sequentially minimizing the length of an interval containing the maximum of a unimodal function on the unit interval without assuming continuity or derivatives. The solution involves a procedure that is ε-minimax among all sequential nonrandomized procedures that terminate by providing an interval containing the required point. The procedure is designed to handle observations with errors, though optimal results for this more challenging case are not yet known. The search for the maximum is described as a "second-order" search, where pairs of observations provide information. The class of functions considered, denoted as \(\mathcal{F}\), includes functions that are either strictly increasing or decreasing on specific intervals. The minimax procedure involves splitting the interval into equal parts at the point of the next observation. The paper defines a strategy \(S\) that consists of a starting point, functions for subsequent observations, and functions for selecting the final interval. The goal is to find a strategy \(S_N^*\) that minimizes the length of the final interval for any function in \(\mathcal{F}\) with \(N\) observations. The procedure \(S_N^*\) is constructed using Fibonacci numbers and specific functions to guide the observations. The proof of the minimax property is given inductively, showing that the procedure satisfies the minimax condition for \(N = 2\) and extending this to larger \(N\). The paper also discusses an alternative procedure \(S^*\) that can be more efficient in certain cases, especially when the number of observations is not fixed in advance. The references include works on stochastic estimation and stochastic approximation methods, which are relevant to the problem of finding the maximum of a function.