grandes-ecoles 2025 Q31

grandes-ecoles · France · centrale-maths2__official Stationary points and optimisation Find critical points and classify extrema of a given function

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
By distinguishing cases according to whether $\beta \leqslant 1$ or $\beta > 1$, show that $G_h$ admits a minimum on $\mathbb{R}_+$ attained at a unique point $u_h$. Show moreover that

[(a)] if $\beta \leqslant 1$ and $h = 0$, then $u_h = 0$;
[(b)] if $\beta > 1$, then $u_h > 0$;
[(c)] for any $\beta \in \mathbb{R}_+^*$,
- [(i)] $G_h'(u_h) = 0$;
- [(ii)] $h = \beta G_0'(u_h)$;
- [(iii)] $G_h''(u_h) > 0$ when $h > 0$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.

We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.

By distinguishing cases according to whether $\beta \leqslant 1$ or $\beta > 1$, show that $G_h$ admits a minimum on $\mathbb{R}_+$ attained at a unique point $u_h$. Show moreover that
\begin{itemize}
\item[(a)] if $\beta \leqslant 1$ and $h = 0$, then $u_h = 0$;
\item[(b)] if $\beta > 1$, then $u_h > 0$;
\item[(c)] for any $\beta \in \mathbb{R}_+^*$,
\begin{itemize}
\item[(i)] $G_h'(u_h) = 0$;
\item[(ii)] $h = \beta G_0'(u_h)$;
\item[(iii)] $G_h''(u_h) > 0$ when $h > 0$.
\end{itemize}
\end{itemize}

Paper Questions

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