 Solution

Put $f(\varphi )=dx/d\varphi$. Since $\sin ((2\pi -\varphi )/2)=\sin (\varphi /2)$, both $f(2\pi -\varphi )=f(\varphi )$ and $y(2\pi -\varphi )=y(\varphi )$. Using $x(2\pi )=1$ in the integral for $x$ givesHence the physical track $(x,-y)$ is symmetric about $x=1/2$, with $x(\pi )=1/2$. For $0<\varphi <2\pi$, $dx/d\varphi >0$, andis positive before $\pi$ and negative after $\pi$. Therefore the physical height $-y$ has its unique minimum at $x=1/2$, withThe depth $y$ has a maximum there. This proves the intended claim for every finite-time member $L>1$; $L=1$ has the degeneracy described in (a).