second derivative test
(1.7 hours to learn)
In an optimization problem, a critical point (where the partial derivatives are zero) may be a local minimum or maximum, or a saddle point. The second derivative test is a way of testing optimality: a point is a (local) minimum if the Hessian matrix is positive definite.
This concept has the prerequisites:
- optimization problems
- higher-order partial derivatives (The test involves second derivatives.)
- positive definite matrices (If the matrix is positive definite, then the point is a local minimum.)
Core resources (read/watch one of the following)
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 5.3, "Quadratic forms and the second derivative test," pages 208-215
Supplemental resources (the following are optional, but you may find them useful)
→ MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
→ Multivariable Calculus
An introductory multivariable calculus textbook.
Location: Section 13.10, "Critical points of functions of two variables," pages 927-933
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