Hessian & Bordered Hessian
Second-order conditions: checking that your optimum is actually a maximum.
Definition
Second-order conditions: the Hessian matrix of second derivatives certifies a critical point as maximum (negative definite) or minimum; the bordered Hessian does the same under constraints.
Key equation
Unconstrained max: leading principal minors alternate sign starting negative
The intuition
First-order conditions find flat points; the Hessian asks which way the surface curls. In economics it's the math behind assuming diminishing marginal utility and convex costs, the conditions that make optima interior and unique.
Exam tip
For two-variable constrained problems, one determinant check of the bordered Hessian settles max vs min, memorize that 3×3 pattern.
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