Minimum-time trajectories for applications where a geometric path is followed by a kinematically redundant robot's end-effector may yield economical improvements in many cases compared to conventional manipulators. While for non-redundant robots the problem of finding such trajectories has been solved, the redundant case has not been treated exhaustively. In this contribution, the problem is treated as two interdependent subproblems: inverse kinematics and trajectory optimization. Therein, a differential inverse kinematics resolution scheme is augmented by adding an optimal linear combination of nullspace basis vectors of the corresponding velocity Jacobian. Using the practical example of an industrial robot with 7 degrees of freedom performing a 5 degrees of freedom task, the effectiveness of the presented method is shown. Comparisons are made with a joint space decomposition inverse kinematics resolution approach.