Final answer:
To determine if a critical point is a local minimum, local maximum, or saddle point in a multivariable function, the Hessian matrix is used. The determinant and trace of the Hessian matrix are used to classify the critical points.
Step-by-step explanation:
When determining whether a critical point of a multivariable function is a local minimum, local maximum, or saddle point, one can test points with the same potential. These points are the critical points of the function where the partial derivatives are zero or undefined. To test these points, the Hessian matrix is used. The Hessian matrix is calculated by taking the second derivative of the function with respect to each variable and arranging them in a matrix.
To classify the critical points, the determinant of the Hessian matrix is computed. If the determinant is positive and the trace (the sum of the diagonal entries) is positive, then the critical point is a local minimum. If the determinant is negative and the trace is negative, the critical point is a local maximum. If the determinant is negative but the trace is positive or the determinant is positive but the trace is negative, the critical point is a saddle point.