The analytical solution boils down to finding the roots of another polynomial. If your fixed point is (x,y) and your polynomial is p(t), then the distance squared is
f(t) = (t-x)^2 +(p(t)-y)^2
Setting the derivative of f(t) to zero leads to
(t-x)+ (p(t)-y)*dp/dt = 0
which is just a problem of finding the roots of the polynomial on the left hand side. Once you find its roots using the ROOTS command, you then test f(t) at the real roots for the minimum distance.
