Multivariate calculus from first principles
The linear approximation (Lesson 9) used only the gradient and gave a flat tangent plane. Add the next term, the one built from the Hessian, and you get a quadratic approximation: a paraboloid that hugs the surface, capturing its curvature, not just its tilt.
Read the three pieces: f(x) is the height, ∇fᵀδ is the linear (slope) correction, and ½δᵀHδ is the quadratic (curvature) correction. That last term is a quadratic form in the step, exactly the object whose sign the Hessian's eigenvalues control.
A flat tangent plane resting on a curved surface is like setting a stiff glass slide on your eye: it touches at one spot but gaps everywhere else. A contact lens does better because it's curved to match the eye's surface, matching not just where the eye is but how it bends. The Hessian term ½δᵀHδ is that built-in curvature: it lets the approximation hug the surface instead of merely resting on it.