We study the effect of the presence of different types of critical points, such as ordinary critical points, multicritical points and quasicritical points, along different paths on the fidelity susceptibility and Loschmidt echo of a three-spin interacting transverse Ising chain using a method that does not involve the language of tensors. We find that the scaling of the fidelity susceptibility and Loschmidt echo with the system size at these special critical points of the model studied is in agreement with the known results, thus supporting our method. © 2014 IOP Publishing Ltd and SISSA Medialab srl.