Jacobiankriteriet

The Jacobian criterion is a mathematical concept used to determine the nature of critical points in multivariable calculus. It is an extension of the second derivative test for single-variable functions. For a function of two variables, say f(x, y), a critical point (a, b) is a point where both partial derivatives, ∂f/∂x and ∂f/∂y, are equal to zero.

To apply the Jacobian criterion, one calculates the second partial derivatives: f_xx, f_yy, and f_xy. These are then used to form the Hessian matrix, whose determinant is known as the Jacobian determinant or simply the discriminant, denoted by D. The formula for D is D = f_xx(a, b) * f_yy(a, b) - [f_xy(a, b)]^2.

The value of D at the critical point (a, b) indicates the nature of that point. If D > 0 and f_xx(a, b) > 0, the point is a local minimum. If D > 0 and f_xx(a, b) < 0, the point is a local maximum. If D < 0, the point is a saddle point, meaning it is neither a local minimum nor a local maximum. If D = 0, the test is inconclusive, and other methods are needed to determine the nature of the critical point. The Jacobian criterion can be generalized to functions of more than two variables using the determinant of the Hessian matrix.