High Quality Content by WIKIPEDIA articles! In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point (x, y) of a function f(x,y) is a minimum, maximum or saddle point. This analysis applies only to a function of two variables. For a function of more variables, one must look at the eigenvalues of the Hessian matrix at the critical point. If all of the eigenvalues are positive then the critical point is a minimum and if all of the eigenvalues are negative then the critical point is a maximum. If some of the eigenvalues are negative and others are positive, the critical point is a saddle point. Finally, if any of the eigenvalues are equal to zero and the rest are either all positive or all negative, then the second derivative test is inconclusive, and some other tool is needed. Note that for functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point. Note also that this statement of the second derivative test for many variables also applies in the two-variable and one-variable case.