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Extended $f(R)$ theories of gravity have been investigated from the symmetrypoint of view. We briefly has been investigated Noether symmetry of two typesof extended $f(R)$ theories: $f(R,T)$ theory, in which curvature is coupled nonminimally to the trace of energy momentum tensor $T_{\mu\nu}$ and mimetic $f(R) $ gravity, a theory with a scalar field degree of freedom, but ghost-freeand with internal conformal symmetry. In both cases we write point -likeLagrangian for flat Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmologicalbackground in the presence of ordinary matter. We have been shown that someclasses of models existed with Noether symmetry in these viable extensions of$f(R)$ gravity. As a motivated idea, we have been investigating the stabilityof the solutions and the bouncing and $\Lambda$CDM models using the Noethersymmetries. We have been shown that in mimetic $f(R)$ gravity bouncing and$\Lambda$CDM solutions are possible. Also a class of solutions with futuresingularities has been investigated.

Cosmological viable Mimetic $f(R)$ and $f(R,T)$ theories via Noether symmetry