One Constraint, One Inequality

Let $f(x) = (a - x)\mathrm{e}^x \ln x$, defined for $x > 0$.

1. If $f(x) \leqslant 0$ for all $x > 0$, find $a$. 2. For this value of $a$, prove that $f(x) < x^a$ for all $x > 0$.