EDB — 1HD

Exercises

1. [1HD] In the same hypotheses, we see a "vice versa". Let \(f,𝜑:A\to {\mathbb {R}}\) be of class \(C^ 2\) in the open set \(A\), and let \(\overline x∈ E_ a\) and \(𝜆∈ℝ\) be such that \(∇ f(\overline x)+𝜆 ∇𝜑(\overline x)=0\); suppose that

   \[ ∀ v, v⋅ ∇ 𝜑(x)=0⟹ v⋅ H v {\gt} 0 \]

   where

   \[ h(x)=f(x)+𝜆𝜑(x) \]

   and \(H\) is the Hessian matrix of \(h\) in \(\overline x\). Show that \(\overline x\) is a local minimum point for \(f\) bound to \(E_ a\).

   Solution 1

   [1HF]↺↻