Extrema of Functions of Two Variables (Hessian Matrix) An extrema of a function of two variables is a point in the domain of the function that maximizes or...
Extrema of Functions of Two Variables (Hessian Matrix)
An extrema of a function of two variables is a point in the domain of the function that maximizes or minimizes the value. For a function of two variables, the second partial derivatives of the function determine the nature of the extrema.
The Hessian matrix is a square matrix of second partial derivatives that gives information about the behavior of the second partial derivatives. The eigenvalues of the Hessian matrix at a critical point indicate the nature of the critical point:
Positive eigenvalues: The critical point is a local minimum.
Negative eigenvalues: The critical point is a local maximum.
Zero eigenvalues: The critical point is a saddle point.
In addition to the eigenvalues, the Hessian matrix also gives information about the second partial derivatives. For example, a positive determinant of the Hessian matrix indicates that the function has a relative minimum at the critical point.
Examples:
The critical point of a function of two variables (f(x, y)) can be found by finding the points where the partial derivatives are equal to zero.
The Hessian matrix of a function of two variables can be used to determine the nature of the critical point.
A positive determinant of the Hessian matrix indicates that the critical point is a local minimum.
A negative determinant of the Hessian matrix indicates that the critical point is a local maximum