How does the Hessian matrix classify stationary points of a twice continuously differentiable two-variable function?

Conditions

• The function is twice continuously differentiable near the point.
• The gradient is zero at the point (stationary point).
• The function has two variables.

Reasoning, step by step

1. Assume the function is twice continuously differentiable and the gradient is zero at the point.
2. Arrange the second partial derivatives into the Hessian matrix.
3. Evaluate the leading second-order change along a small displacement using the quadratic form.
4. Determine the definiteness of the Hessian matrix.
5. Classify the point based on the definiteness: positive definite for minimum, negative definite for maximum, indefinite for saddle, semidefinite for inconclusive.

Example

For a function with Hessian $H$, the leading second-order change is $\frac12 h^T Hh$. If $H$ is positive definite, this change is positive in every nonzero direction, indicating a strict local minimum.

Common misconceptions

• Believing that a positive definite Hessian alone guarantees a global minimum without checking the gradient.
• Assuming that a semidefinite Hessian always indicates a saddle point.
• Thinking that the Hessian classification works for points where the gradient is not zero.

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Connected concepts

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