Cosine Versus Exponential

Let $f(x) = ax^2 - \mathrm{e}^{x-1}$.

1. When $a = \dfrac{1}{2}$, prove that $f(x)$ is decreasing on $\mathbb{R}$. 2. If $f(x) \leqslant a\cos x$ for all $x \in \left[0, \dfrac{\pi}{2}\right]$, find the range of possible values of the real number $a$.