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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

$\displaystyle f'(x)$ $\textstyle =$ $\displaystyle 3x^2$  
$\displaystyle f''(x)$ $\textstyle =$ $\displaystyle 6x$  
$\displaystyle f'''(x)$ $\textstyle =$ $\displaystyle 6.$  

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.

© 1996-9 Eric W. Weisstein