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For this I started with the 1st interval [0,pi/2] and said

arctan [4/ ... ] = [0 , pi/2]

but studying the tan graph this is all positive values of the y-axis

so surely the range of values for this is (0 , inf)?

If I've done that right, how do I do the other interval [-pi/2 , pi]?

Because using that method gives the result (-inf , 0), but then that would also be true for the quadrant [0 , -pi/2] surely, which brings me back to my first answer also being true for [pi/2 , pi]..

How do I attempt this because this is obviously wrong..

This was under the chapter about Laplace / Fourier Transforms

And if you could shed any light on how to maximise too I'd be greatful, because my notes havn't touched on this..

The y(t) = ... was formed from solving a differential equation using laplace transforms, I didn't include that because I didn't think it was relevant. =)