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[1F6]Let now \(A β β^ n\) an open non-empty set, let \(f,π:A\to {\mathbb {R}}\) be real functions of class \(C^ 1\) on \(A\). Having fixed \(aββ\) we then define the level set
\[ E_ a =\{ xβ A : π(x) = a\} \]
we assume that \(E_ a\) is non-empty, and that \(β π(x) β 0\) for each \(x β E_ a\).
We call local minimum point of \(f\) bound to \(E_ a\) a point of \(E_ a\) that is a local minimum for \(f|_{E_ a}\); and similarly for maxima.