Practice question · Multiple choice
Definiteness of a quadratic form and the second-order condition for an optimum are the same test. Which correspondence makes that true?
Hints
- Take the second-order Taylor expansion at a critical point. What object appears?
- Ask what 'curves downward in every direction' means as a statement about a matrix.
Show the answer
D. The Hessian's definiteness IS the local curvature the SOC tests
Why
The first-order term vanishes at a critical point, so the local shape is entirely the quadratic form built from the Hessian, definiteness is the curvature condition. Option 2 confuses the second-order term with the whole approximation, which also carries the function's value.
Practise Quadratic Forms
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