EDB — 1GB

[1GB] Prerequisites:[1G8]↺↻.Let \(V,W⊆ ℝ^ n\) be open nonempty sets, and \(G:V→ W\) of class \(C^ 2\). Fix \(\overline y∈ V\) and \(\overline x=G(\overline y)∈ W\). Suppose that \(𝜓:W→ℝ\) is of class \(C^ 2\); define \(\tilde𝜓 = 𝜓 ◦ G\), then compare Taylor’s second-order formulas for \(𝜓\) and \(\tilde𝜓\) (centered in \(\overline x\) and \(\overline y\), respectively). Assuming also that \(G\) is a diffeomorphism, verify that

• \(\overline x\) is a stationary point for \(𝜓\) if and only if \(\overline y\) is stationary point for \(\tilde𝜓\),

• and in this case the Hessians of \(𝜓\) and \(\tilde𝜓\) are similar (i.e. the matrices are equal, up to coordinate changes).