For a multi-input function , the critical point of the function is where its derivative/gradient is zero or when the derivative/gradient doesn’t exist:

Critical point can be view as a generalization of extremum for single-variable function, but it does not necessary be an extremum. Instead, it may either be a maximum, minimum, or a saddle point.

min max and saddle.png

We can use the second derivative test to classify critical points.

Critical point at where the Derivative Doesn’t Exist

The classic example of a critical point at where the derivative doesn’t exist is