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This paper addresses the problem of determining the feasible speeds and voltages of multicore processors with hard real-time and temperature constraints. This is an important problem, which has applications in time-critical execution of programs like audio and video encoding on application-specific embedded processors. Two problems are solved. The first is the computation of the optimal time-varying voltages and speeds of each core in a heterogeneous multicore processor, that minimize the makespan-the latest completion time of all tasks, while satisfying timing and temperature constraints. The solution to the makespan minimization problem is then extended to the problem of determining the feasible speeds and voltages that satisfy task deadlines. The methods presented in this paper also provide a theoretical basis and analytical relations between speed, voltage, power and temperature, which provide greater insight into the early-phase design of processors and are also useful for online dynamic thermal management.