Saddle Point (Function)

A Point of a Function or Surface which is a Stationary Point but not an Extremum. An example of a 1-D Function with a saddle point is $f(x)=x^3$, which has

$$f'(x) = 3x^2$$

$$f''(x) = 6x$$

$$f'''(x) = 6.$$

This function has a saddle point at $x_0=0$ by the Extremum Test since $f''(x_0)=0$ and $f'''(x_0)=6\not=0$. An example of a Surface with a saddle point is the Monkey Saddle.