second-order Taylor expansion (several variables)
A first-order approximation replaces a surface near a point by its tangent plane — flat, no curvature. That is often too crude: a tangent plane cannot tell a valley from a hilltop from a saddle. The second-order Taylor expansion adds the curvature term, fitting the best bowl-shaped (quadratic) surface to the function near the point so that not just the value and slope but also the bending match.
Written compactly, near a point a, f(a + h) is approximately f(a) + grad f(a) dot h + (1/2) h^T H(a) h, where grad f(a) is the gradient (the linear, tilting part) and H(a) is the Hessian matrix (the quadratic, curving part), and h^T H h is the quadratic form that reads off the curvature in the direction of the step h. The three pieces are: the value at the base point, the linear correction (the total differential), and the quadratic correction. The leftover error is third order in |h|, so this approximation is markedly better than the linear one for small steps. It is the multivariable analogue of f(a) + f'(a) h + (1/2) f''(a) h^2.
This expansion is the bridge from the Hessian to actual conclusions. At a critical point the gradient term vanishes, leaving f(a + h) approximately f(a) + (1/2) h^T H h, so the local shape is governed entirely by the Hessian's quadratic form — positive definite gives a minimum, negative definite a maximum, indefinite a saddle. That is precisely the second-derivative test. The same quadratic model is what Newton's method minimizes at each step, what defines the curvature of an energy landscape near equilibrium, and what underlies error analysis and uncertainty quantification when first-order terms cancel.
Expand f(x, y) = e^x cos(y) about (0, 0). Here f(0, 0) = 1, grad f = (1, 0), Hessian [1, 0; 0, -1], so f is approximately 1 + x + (1/2)(x^2 - y^2) for small (x, y) — value, tilt, and curvature all matched.
Value plus gradient term plus one-half the Hessian quadratic form — the quadratic that hugs the surface.
When the Hessian is degenerate (a zero eigenvalue) at a critical point, the quadratic term cannot decide the local shape, and you must go to third- or higher-order terms — the second-order test simply returns 'inconclusive'.