In Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AE-extensions, have been defined. They provide the same interpretations as the extensions to modal intervals and therefore enhance the interpretations of the classical interval extensions (for example, both inner and outer approximations of function ranges are in the scope of the AE-extensions). The construction of AE-extensions is similar to the one of classical interval extensions. In particular, a natural AE-extension has been defined from the Kaucher arithmetic which simplified some central results of the modal intervals theory. Starting from this framework, the mean-value AE-extension is now defined. It represents a new way to linearize a real function, which is compatible with both inner and outer approximations of its range. With a quadratic order of convergence for real-valued functions, it allows to overcome some difficulties which were encountered using a preconditioning process together with the natural AE-extensions. Some application examples are finally presented, displaying the application potential of the mean-value AE-extension.