The classical Brachistochrone was actually counterintuitive wherein the time of descent is lesser (the least) for the cycloid than that of the corresponding straight inclined track.
Let an inclined and wavy track be given by $$y= 1-xe^{\epsilon \cos^2(7x\pi/2)}~~~(1)$$ for $\epsilon=0$ this is a simple inclined, straight track and the time taken by for a particle to fall down is $2 \tau$, where $\tau=\frac{1}{\sqrt{g}}$ in CGS.
The question is on which track the time of descent will be lesser: on the red ($\epsilon=+0.06$) or on the blue ($\epsilon=-0.06$) track? Justify your answer with or without a calculation. See the figure below for the two complementary inclined tracks.