Saddle Points and Turning Points

Consider the point $(0,0)$ on the graphs of $y=x^2$ and $y=x^3$. Clearly, it is a stationary point on both graphs, but there is a very clear difference between the two. The graph of $y=x^2$ "turns around" at this point, while the graph of $y=x^3$ merely "takes a quick break" before going in the same direction in which it was already going.

Thus, we see that there are two types of stationary points.

1. Local Extremum

Obviously, one type of stationary point is a local extremum. Here, the value of the function is either greater than its nearby values (relative maximum) or less than its nearby values (relative minimum).

1. Saddle Point

At a saddle point, the function has neither a minimum nor a maximum. For a function $f(x)$, a saddle point (or point of inflection) is any point at which $f(x)$ is continuous and $f''(x)$ changes sign, which means that $f(x)$ changes concavity.

Thus, we can classify the stationary points, the points where $f'(x)=0$, by using the value of $f''(x)$.

If $f''(x)>0$, then $f$ has a relative minimum at $x$.

If $f''(x)<0$, then $f$ has a relative maximum at $x$.

If $f''(x)=0$, and $f''(x)$ changes sign, then $f$ has a saddle point at $x$.

If $f''(x)=0$, $f''(x)$ does not change sign, and $f''>0$ for values very close to $x$, then $f$ has a relative minimum at $x$.

If $f''(x)=0$, $f''(x)$ does not change sign, and $f''<0$ for values very close to $x$, then $f$ has a relative maximum at $x$.