It was during the solving of this puzzle on Wolfram Alpha that I decided to subscribe to Wolfram Alpha Pro.
Wolfram Alpha gives the answer as sqrt(109), at x = 300/763  3*sqrt(4441)/763 or (300  3sqrt(4441))/763 ~= .131163191378042.
The derivative of the function given in the puzzle was found to be f'(x) = (70 x)/(sqrt(1  x^2) sqrt(149  140 sqrt(1  x^2)))  21/sqrt(58  42 x), which has a zero at the above mentioned value of x.
This is from Wolfram's text version of the stepbystep solution.

Posted by Charlie
on 20190216 15:11:11 