Multivariate calculus from first principles
The Hessian's eigenvalues turn the murky question 'what kind of critical point is this?' into a clean checklist. At a point where the gradient is zero, the signs of the Hessian's eigenvalues tell you whether you're sitting in a bowl, on a dome, or at a saddle.
This is the multivariable second-derivative test, and it's a direct generalization of 1-D: there, f″ > 0 meant a min and f″ a max. The Hessian's eigenvalues are the many directions' versions of that single number.
Picture three snacks. A bowl of soup curves up no matter which way you tip it, a dome of ice cream curves down everywhere, and a Pringle chip bends up along its length but down across its width. The Hessian's eigenvalues are just the curvatures along those special directions: same sign means bowl or dome, opposite signs (like 2 and −2) means the chip, a saddle.