A Double-Root Cubic

Let $f(x) = (x - 1)^2(x - 4)$. Determine, with proof, which of the following are true:

1. $x = 1$ is a local minimum point of $f(x)$; 2. $f(2 + x) + f(2 - x) = -4$ for all $x$; 3. the solution set of $-4 < f(2x - 1) < 0$ is $\{x \mid 1 < x < 2\}$; 4. for $0 < x < \dfrac{\pi}{2}$, $f(\sin x) > f\left(\sin^2 x\right)$.