A Nonnegative Minimum

Let $a > 0$ and $f(x) = (1 - ax)\left(\mathrm{e}^x - 1\right)$.

1. If $a = 1$, prove that $f(x) < \ln(x+1)$ for $x > 0$. 2. If $h(x) = \ln(x+1) - f(x)$ has a local minimum point $x_0$, prove that $f(x_0) \geqslant 0$.