The onset of double‐diffusive convection is studied for an infinite fluid layer possessing a density maximum in its interior. Linear stability analyses are performed on a basic state with constant temperature and concentration gradients in which the density‐temperature relationship is quadratic. Regions of stability and instability to both steady and oscillatory modes are delineated in the positive quarterplane of the Rayleigh and solutal Rayleigh numbers plane. It is found that decreasing the position of maximum density in the vertical leads to an increased stability range and also increased regions of oscillatory instability in that quadrant. It is also found that the extent of penetration of the convective motion into the stable region is diminished with increased solute concentration. The effects of both the rigid–rigid and the free–free boundary conditions on the instabilities are investigated. Both conditions lead to dynamically similar motions with significant quantitative differences.