Extremum of a function obtained by ordering two specified smooth functions (Q1897216)

Let us given smooth real-valued functions \(f^1\) and \(f^2\) in a neighborhood of the point \(x^0\) in the \(n\)-dimensional Euclidean space. Set \(F^1 (x) = \min \{f^1 (x), f^2 (x)\}\) and \(F^2(x) = \max \{f^1 (x),\;f^2 (x)\}\). The author derives necessary and sufficient conditions for the functions \(F^1\) and \(F^2\) to have an extremum at the point \(x^0\). There is a misprint in the item 2 of the references: the name ``Dement'ev'' must be replaced by ``Dem'yanov''.