(a) Let $a$ and $b$ be in $[0, +\infty[$ such that $a < b$. Show that there exists $\epsilon \in ]0, b]$ such that $\varphi(a + t) + \varphi(b - t) > \varphi(a) + \varphi(b)$ for all $t > 0$ such that $t \leqslant \epsilon$. (b) Deduce that $H_{N}$ attains its maximum on $\Sigma_{N}$ at a unique point which you will determine. Where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise,} \end{cases}$ $\Sigma_{N}$ is the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$, and $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$.
(a) Let $a$ and $b$ be in $[0, +\infty[$ such that $a < b$. Show that there exists $\epsilon \in ]0, b]$ such that $\varphi(a + t) + \varphi(b - t) > \varphi(a) + \varphi(b)$ for all $t > 0$ such that $t \leqslant \epsilon$.
(b) Deduce that $H_{N}$ attains its maximum on $\Sigma_{N}$ at a unique point which you will determine.
Where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise,} \end{cases}$ $\Sigma_{N}$ is the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$, and $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$.