Find dL/dx set to zero. Solve for x.
(27+x)/x * sqrt(64+x^2)
What a nasty function to differentiate!
I wrote it as
L = [(27+x)√(64+x^2)]/x
used a combination of quotient rule and product rule and got an answer of
dL/dx = (x^3-1728)/[(x^2)(√(64+x^2)]
setting this to zero gives only one real result:
x^3 = 1728
x = 12