  From a recent geometric generalization of Thermodynamic Uncertainty Relations
(TURs) we derive novel upper bounds on the nonlinear response of an observable
of an arbitrary system undergoing a change of probabilistic state. Various
relaxations of these bounds allow to recover well known bounds such as
(strengthenings of) Cramer-Rao's and Pinsker's inequalities. In particular we
obtain a master inequality, named Symmetric Response Intensity Relation, which
recovers several TURs as particular cases. We employ this set of bounds for
three physical applications. First, we derive a trade-off between thermodynamic
cost (dissipated free energy) and reliability of systems switching instantly
between two states, such as one-bit memories. We derive in particular a lower
bound of $2.8 k_BT$ per Shannon bit to write a bit in such a memory, a bound
distinct from Landauer's one. Second, we obtain a new family of classic speed
limits which provide lower bounds for non-autonomous Markov processes on the
time needed to transition between two probabilistic states in terms of a
thermodynamic quantity (e.g. non-equilibrium free energy) and a kinetic
quantity (e.g. dynamical activity). Third, we provide an upper bound on the
nonlinear response of a system based solely on the `complexity' of the system
(which we relate to a high entropy and non-uniformity of the probabilities). We
find that `complex' models (e.g. with many states) are necessarily fragile to
some perturbations, while simple systems are robust, in that they display a low
response to arbitrary perturbations.
