The resulting Hessian is
The Hessian matrix will be symmetric if the partial derivatives of the function are continuous.
The determinant of a Hessian matrix can be used as a generalisation of the second derivative test for single-variable functions. If the determinant of the Hessian positive, it will be an extreme value (minimum if the matrix is positive definite). If it is negative, there will be a saddle point. If it is 0, another test must be used.
A bordered Hessian is a similar matrix used to optimize a multivariable function with a constraint .