Extrema of a function of two variables

For a two-variable function, define $D=f_{xx}f_{yy}-f_{xy}^2$. If $D>0$, the sign of $f_{xx}$ distinguishes the type: $f_{xx}>0$ indicates a local minimum, and $f_{xx}<0$ indicates a local maximum.

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

A semidefinite Hessian is inconclusive because the second-order term does not strictly determine the sign of the change in all directions. Higher-order terms or the function's behavior directly must be examined to determine if the point is a minimum, maximum, or saddle.

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

The Hessian matrix classifies stationary points by analyzing the sign of the second-order change along a small displacement. If the Hessian is positive definite, the point is a strict local minimum.

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