Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying:
$$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$
\textbf{VI.A.1)} Let $f$ be a function of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{4}$. Show that $f$ admits a maximum on $\Omega$. We then denote $a = (a_{1}, a_{2}, a_{3}, a_{4}) \in \Omega$ a point at which this maximum is attained.
\textbf{VI.A.2)} Show that if $(x_{1}, x_{2}, x_{3}, x_{4}) \in \Omega$ then $x_{3}$ and $x_{4}$ can be written in the form
$$\begin{aligned} & x_{3} = u x_{1} + v x_{2} + w \\ & x_{4} = u^{\prime} x_{1} + v^{\prime} x_{2} + w^{\prime} \end{aligned}$$
where we shall give explicitly $u, v, u^{\prime}, v^{\prime}$ in terms of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$.
\textbf{VI.A.3)} Assuming that none of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ is zero, deduce that
$$\begin{aligned} & \frac{\partial f}{\partial x_{1}}(a) + u \frac{\partial f}{\partial x_{3}}(a) + u^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \\ & \frac{\partial f}{\partial x_{2}}(a) + v \frac{\partial f}{\partial x_{3}}(a) + v^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \end{aligned}$$
\textbf{VI.A.4)} Show that the vector subspace of $\mathbb{R}^{4}$ spanned by the vectors $(1, 0, u, u^{\prime})$ and $(0, 1, v, v^{\prime})$ admits a supplementary orthogonal subspace spanned by the vectors $(1,1,1,1)$ and $(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4})$.
\textbf{VI.A.5)} Deduce the existence of two real numbers $\alpha, \beta$ such that for all $i \in \{1,2,3,4\}$ we have
$$\frac{\partial f}{\partial x_{i}}(a) = \alpha + \beta \varepsilon_{i}$$