EDB — 1F6

[1F6]Let now \(A ⊆ ℝ^ n\) an open non-empty set, let \(f,𝜑:A\to {\mathbb {R}}\) be real functions of class \(C^ 1\) on \(A\). Having fixed \(a∈ℝ\) we then define the level set

we assume that \(E_ a\) is non-empty, and that \(∇ 𝜑(x) ≠ 0\) for each \(x ∈ E_ a\).

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.