grandes-ecoles 2025 Q32

grandes-ecoles · France · centrale-maths2__official Stationary points and optimisation Find absolute extrema on a closed interval or domain
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
We now assume that $h > 0$. Justify that $G_h$ attains its minimum on $\mathbb{R}$ at a unique point which is $u_h$. 
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.

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

We now assume that $h > 0$. Justify that $G_h$ attains its minimum on $\mathbb{R}$ at a unique point which is $u_h$.
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