A linear stability analysis of Rayleigh–Bénard convection in a Newtonian nanofluid is carried out using most general boundary conditions. A single-phase description of nanofluids is adopted in the study. The nanofluids used for the study are water–alumina and water–copper nanofluids in order to analyze how a choice between them can be made. The values of thermophysical quantities of nanofluids are calculated using the mixture theory and phenomenological-laws. The paper applies the Maclaurin series in solving the boundary-eigenvalue-problem through a simple and innovative approach. A single-term Galerkin technique is adopted to obtain the guess value of the critical Rayleigh number and the wave number. Further, improved values of the Rayleigh number and the wave number are obtained using the solution of a system of three linear-algebraic equations. A detailed discussion is made on the effect of rough-boundaries and Robin-boundary conditions for temperature on the onset of convection. A comparative study between the results of two nanofluids is made and the destabilizing effect of nanoparticles in the Newtonian carrier-fluid on the onset of convection is studied.