MonkeySaddle

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 f(x,y) = x^3--3*x*y^2. The gradient of this function is . The Hessian determinant is -36x^2 - 36y^2 which evaluates to zero at the origin (which is where the lone critical point is located).