Pinning a Sine Wave

Let

and suppose the line $y = a$ meets the graph of $f$ at consecutive intersection points with $x$-coordinates $2$ and $4$. Determine, with proof, which of the following are true:

1. on $[2, 4]$ the range of $f$ is $\left[a, \sqrt{2}a\right]$; 2. on $[2, 4]$ the function attains its maximum at $x = 3$; 3. the smallest positive period can be $\dfrac{8}{3}$; 4. the graph of $f(x)$ can pass through the origin.