the 6th derivative, which is positive. Thus n = 5, and by the test, there is a local minimum at 0. Multivariable case For a function of more than one variable, the second-derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point. In particular, assuming that all second-order partial derivatives of f are continuous on a neighbourhood of a critical point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum. If the eigenvalues are all negative, then x is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second-derivative test is inconclusive. See also Fermat's theorem (stationary points) Maxima and minima Karush–Kuhn–Tucker conditions Phase line – virtually identical diagram, used in the study of ordinary differential equations Bordered Hessian Optimization (mathematics) Differentiability Convex function Second partial derivative test Saddle point Inflection point Stationary point Further reading References External links "Second Derivative Test" at Mathworld Concavity and the Second Derivative Test Thomas Simpson's use of Second Derivative Test to Find Maxima and Minima at Convergence Differential calculus