Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are aiming at the point $(1,1)$. Assume the smooth curve of interest has parametric equations $(x(t),y(t))$. Then the curve has continuous derivatives in its parametric interval and $\dot x^2+\dot y^2\not=0$. Without further due, the notes I am reading claim that under our assumption for smoothness a unique implicit function $y(x)$ exists and $y_x>0$.
Our physical intuition agrees with those claims. How can we see analytically that indeed an implicit function $y(x)$ exist and not $x(y)$, and that the derivative $y_x>0$? In case additional assumptions are needed which would be the minimal set of sufficient additional assumptions?
p.s. I can give more details from the notes but I believe I have considered all the relevant assumptions.