Construction of a confidence set for a maximum point of a function is an important statistical problem which has many applications. In this article, an exact 1 − α confidence set is provided for a maximum point of a univariate polynomial function in a given interval. It is shown how the construction method can readily be applied to many parametric and semiparametric regression models involving a univariate polynomial function. Examples are given to illustrate this confidence set and to demonstrate that it can be substantially narrower and so better than the only other confidence set available in the statistical literature that guarantees 1 − α confidence level.