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This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor--Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio 0.0556, the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) where the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible, (2) the perturbed flow is three dimensional and incompressible, (3) the perturbed flow is two dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor--Couette flow, and find that the analytical result indeed agrees well with the available DNS data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.