I didn't check your final result, I can only show you how to answer the last question:
You've got a function with absolute values. Change it into a piece-wise defined function:
As you can see you've got 3 pieces of straight lines which are connected.
The change of the slope from negative to positive indicates a minimum. This happens at h = -0.75
I've attached a drawing of the graph of your function.
A Happy New Year!