Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
It is admitted that, for all real $a$ in the interval $]0; 1]$, the area of triangle $\mathrm{O}N_aP_a$ in square units is given by $\mathscr{A}(a) = \frac{1}{2}a(1 - \ln a)^2$.
Using the previous questions, determine for which value of $a$ the area $\mathscr{A}(a)$ is maximum. Determine this maximum area.