Practice question · Multiple choice
A negative definite Hessian at a critical point identifies a maximum. Why does the sign pattern settle it rather than the determinant alone?
Hints
- Ask what the quadratic form measures in an arbitrary direction.
- What does a saddle point look like along two different directions?
Show the answer
A. Because negative definiteness means it curves down every way
Why
The eigenvalues are curvatures along the principal axes, and their signs decide the classification. A determinant can be positive with two negative eigenvalues, which is why it is not enough on its own.
Practise Quadratic Forms
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