The present article deals with some new features of static anisotropic relativistic compact object with two fluids distribution (strange quark matter (SQM) and normal baryonic matter) along with Karori–Barua (KB) type metric within the framework of so called f(R) theory of gravity. We have considered the MIT bag model equation of state (EoS) to describe the relationship between pressure and matter density for the quark matter distribution inside the celestial object. For this investigation, we have formulated the field equations in f(R) theory with MIT bag EoS together with KB ansatz in the background of two specific viable f(R) models (f(R)=R+αR2 and f(R)=R+α(R2+ζR3)), respectively. Further, the physical credibility of the obtained solutions have been analyzed in the form of matter variables, energy conditions, hydrostatic equilibrium equation and stability criterion by using the graphical representation for different choices of free parameter α. Moreover, we have evaluated that the matter density and radial pressure are central singularity free and remain finite throughout the interior of the fluid sphere. The numerical values of such parameters and bag constant have been calculated corresponding to different compact star candidates (SAXJ1808.4-3658 (x=7.07Km), 4U1820-30 (x=10Km) and PSRJ1614-2230 (x=10.3Km)). Finally, we have observed that our proposed compact star candidate with two viable f(R) models, satisfies all the necessary and sufficient conditions for potentially stable configuration, which are quite suitable to test the strange star candidates.