In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point. The test Functions of two variables Suppose that is a differentiable real function of two variables whose second partial derivatives exist and are continuous. The Hessian matrix of is the 2 × 2 matrix of partial derivatives of : Define to be the determinant of . Finally, suppose that is a critical point of , that is, that ). Then the second partial derivative test asserts the following: If and then is a local minimum of . If and then is a local maximum of . If then is a saddle point of . If then the point could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that is positive at implies that and have the same sign there. Therefore, the second condition, that be greater (or less) than zero, could equivalently be that or be greater (or less) than zero at that point. A condition implicit in the statement of the test is that if or , it must be the case that and therefore only cases 3 or 4 are possible. Functions of many variables For a function f of three or more variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate). In those cases not listed above, the test is inconclusive. For functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point, because the number of jointly sufficient