= Solution
For a <free particle>, vary the <Euclidean action> with a variation $h$ obeying $h(0)=h(T_E)=0$. <Integration by parts> gives
$$
\delta S_{\mathrm{Euc}}=m\int_0^{T_E}\dot x\dot h\,d\tau
=-m\int_0^{T_E}\ddot xh\,d\tau.
$$
The <Euler-Lagrange equation> is therefore $\ddot x_0=0$. The fixed <boundary conditions> determine its unique solution:
$$
\boxed{x_0(\tau)=x_i+\frac{x_f-x_i}{\beta\hbar}\tau,
\qquad 0\leq\tau\leq\beta\hbar.}
$$
This <classical path> is also the unique minimum among paths with finite <Euclidean action>: the second variation is $m\int_0^{T_E}\dot h^2d\tau>0$ for every nonzero admissible variation. Its <Euclidean action> is
$$
\boxed{S_{\mathrm{Euc}}[x_0]=\frac{m(x_f-x_i)^2}{2\beta\hbar}.}
$$
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