What is the determinant criterion for classifying stationary points of a two-variable function using the Hessian?

Conditions

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

Reasoning, step by step

1. Calculate the second partial derivatives $f_{xx}$, $f_{yy}$, and $f_{xy}$.
2. Compute the determinant $D=f_{xx}f_{yy}-f_{xy}^2$.
3. If $D>0$, check the sign of $f_{xx}$: positive for minimum, negative for maximum.
4. If $D<0$, classify the point as a saddle.
5. If $D=0$, conclude that the test is inconclusive.

Example

If $D>0$ and $f_{xx}>0$, the Hessian is positive definite, giving a strict local minimum. If $D<0$, the Hessian is indefinite, giving a saddle.

Common misconceptions

• Believing that $D=0$ always means the point is a saddle.
• Forgetting to check the sign of $f_{xx}$ when $D>0$.
• Applying this criterion to points where the gradient is not zero.

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Connected concepts

Explore next

• Why does $D<0$ imply a saddle point?
• What should be done if $D=0$?
• How does this relate to the eigenvalues of the Hessian?