Why is a semidefinite Hessian inconclusive for classifying stationary points?

Conditions

• The function is twice continuously differentiable.
• The gradient is zero at the point.
• The Hessian is semidefinite (positive or negative semi-definite).

Reasoning, step by step

1. Recognize that a semidefinite Hessian has zero eigenvalues.
2. Understand that the second-order change is zero in some directions.
3. Realize that the sign of the function change in those directions depends on higher-order terms.
4. Conclude that the second-order test alone cannot classify the point.

Example

If the Hessian is positive semidefinite, the second-order change is non-negative, but it is zero in some directions. The actual behavior in those flat directions requires looking at third or fourth-order derivatives.

Common misconceptions

• Believing that a positive semidefinite Hessian guarantees a local minimum.
• Assuming that a negative semidefinite Hessian guarantees a local maximum.
• Thinking that the Hessian test is always sufficient for classification.

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Connected concepts

Explore next

• What tools can be used when the Hessian test is inconclusive?
• How do higher-order derivatives affect the classification?
• Can you give an example of a function with a semidefinite Hessian at a stationary point?