The integration of machine learning into reactive chemistry, materials discovery, and drug design is transforming the development of novel molecules and materials. Machine Learning Interatomic Potentials (MLIPs) predict potential energies and forces with quantum chemistry accuracy, surpassing traditional approaches. Incorporating force fitting in MLIP training enhances potential-energy surface predictions and improves model transferability and reliability. This paper introduces and evaluates the integration of Hessian matrix training in MLIPs, which encodes second-order information about the PES curvature. Our evaluation focuses on models trained only to equilibrium geometries and first-order saddle points (i.e., critical points on the potential surface), demonstrating their ability to extrapolate to nonequilibrium geometries. This integration improves extrapolation capabilities, allowing MLIPs to accurately predict energies, forces, and Hessian predictions for nonequilibrium geometries. Hessian-trained MLIPs enhance reaction pathway modeling, transition state identification, and vibrational spectra predictions, benefiting molecular dynamics (MD) simulations and Nudged Elastic Band (NEB) calculations. By analyzing models trained with varying combinations of energy, force, and Hessian data on a small molecule reactive data set, we demonstrate that models including Hessian information not only extrapolate more accurately to unseen molecular systems, improving accuracy in reaction modeling and vibrational analysis, but also reduce the total amount of data required for effective training. However, the primary trade-off is increased computational expense, as Hessian training requires more resources than conventional energy-force training. Our findings provide comprehensive insights into the advantages and limitations of Hessian integration in MLIP training, allowing practitioners in computational chemistry to make informed decisions about employing this method in accordance with their specific research objectives and computational constraints.