Zeros of a Derivative

Let $f(x) = \mathrm{e}^{mx} + x - x\ln x$ with $m \geqslant 0$.

1. When $m = 1$, find the range of $f(x)$ on $[1, \mathrm{e}]$. 2. Let $f'(x)$ be the derivative of $f(x)$. Determine the number of zeros of $f'(x)$, discussing all cases.