Suppose that $C$ is a caustic curve with respect to a regular boundary parametrized by $\gamma(t)$. Fix a point $A\in C$, take a tangent ray to $C$ in $A$, intersect it with $\gamma$ (say in $\gamma(T)$), take the reflected ray from $\gamma(t)$, and call $B$ the intersection of the reflected ray with $C$ (the reflected ray is again tangent to $C$ in some point $B$ by definition of caustic curve).

Now let $A$ and $B$ be fixed, and consider the function $$F(t)=d(A,\gamma(t))+d(B,\gamma(t)).$$

I must prove that not only $F'(T)=0$ (which I know since $T$ is a local minimum for $F$) but also $F''(T)=0$. Here the presence of the caustic curve must enter. But why is this true?

Thank you in advance.


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