So we need the minimum value of i.e.
Sorry, I've no more time at present to investigate further.
Edit: added later
This does indeed give a solution. The value of the expression must be positive, which means or . If you differentiate, and put the result equal to zero, you get a quadratic in , which gives values of in the permissible ranges of
Substituting either of these values back gives the minimum value of as exactly , but there's a lot of manipulation of surds along the way. (I've checked this numerically on a spreadsheet, and am pretty sure this is correct.)