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 $\bar{x}\in E_a$ and $𝜆\in ℝ$ be such that $\nabla f(\bar{x})+𝜆\nabla 𝜑(\bar{x})=0$; suppose that

   $$\forall v,v\cdot \nabla 𝜑(x)=0⟹v\cdot Hv>0$$

   where

   $$h(x)=f(x)+𝜆𝜑(x)$$

   and $H$ is the Hessian matrix of $h$ in $\bar{x}$. Show that $\bar{x}$ is a local minimum point for $f$ bound to $E_a$.

   Solution 1

   [1HF]↺↻