The calculation of the external work expended during the switching between two systems yields hounds un their free energy difference. The quality of these bounds depends on the choice of path along which the work is calculated. In this paper, it is shown that for sufficiently fast relaxations rile optimal path will be characterized by a minimal thermodynamic length, and that it should be parametrized such that it is traversed with constant thermodynamic speed. As an example, the optimal switching of a magnetic field containing a paramagnet is considered, Finally, the effects of finite relaxation rimes are discussed.