A classic example of a function of 2 variables having a critical point (where the gradient is 0) that is neither a maximum nor a minimum. At the critical point the Hessian determinant is zero, thus the multi-variable version of the 2nd derivative test is inconclusive.

The equation is . The gradient of this function is . The Hessian determinant is which evaluates to zero at the origin (which is where the lone critical point is located).