Given f(x) = x^a/(x^b +1), Find the x coordinate(s) of the horizontal tangent line(x) on f's graph

Quotient Rule of Calculus and some slightly challenging algebra(tougher than the concomitant product rule algebra regarding factoring) were used to express the x coordinate of the horizontal tangent line in terms of the "parameters" a and b in the exponential quotient. The derivative quotient only had one factor that could be equal to zero. Thank you to Wrath of Math for his video on impossibly difficult Calculus exam questions back in 2008. This modified problem probably came from a take home exam due to the exceedingly tedious and error prone problems. Original ill-posed problem was how many horizontal tangent lines does a quotient in this form possess? There was long highly critical thread of commentary at Reddit.
BlacktshirtMathProfessor covered the rudiments of this topic June 2021. New Jersey native Mathematics professor Jonathan Flint February 2012 presented related content.
Find Parameterized XCoordinates Of Horizontal Tangent Line On Exponential Quotient Graph