Does the stability of f(R) theories imply the stability of the dual scalar-tensor theory ?

Higher curvature f(R) gravity theories are often plagued with Ostragadsky instability. In this work we show that such instability manifests itself in the corresponding dual scalar tensor theory in the scalar sector Lagrangian. We explicitly demonstrate the correspondence between the instabilities that appear in an $f(R)$ model and its corresponding scalar tensor theory. Considering various forms of f(R) gravity this feature is illustrated for different choices of the parameters of the theory.