Hello nh149I don't know whether this works, but have you tried the substitution , since the expressions are homogeneous?

Then

and becomes

So we need the minimum value of i.e.

Sorry, I've no more time at present to investigate further.

Grandad

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

and

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.)

Grandad