Let $f \in \mathbb{R}^{N}$ and $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$. We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$.
Show that $F$ is differentiable and calculate its derivative $F'$. Show further that for all $\beta \in ]0, +\infty[$, there exists $p(\beta) \in \Sigma_{N}(\beta f)$ such that $F'(\beta) = -\frac{1}{\beta^{2}} H_{N}(p(\beta))$.