# What's the difference between saddle and inflection point?

As far as I know, inflection point is a point where the concavity of the function changes. These points can be found by taking the second derivative and check if it is 0 and see whether the concavity actually changes on the either sides of the point.

I saw a video which states that saddle point is essentially the same thing but it applies to functions having two or more input variables. But in this link http://mathworld.wolfram.com/SaddlePoint.html, they have found saddle points of a single variable function.

I am confused between both of these.

• I'm not sure the Wolfram definition for saddle point is standard. It's not even really consistent--compare its definition in 1-d to 2-d.
– Paul
Sep 26, 2017 at 19:23
• This site has a clear diagram: quora.com/… May 19, 2023 at 6:17

A point of a function or surface which is a stationary point but not an extremum.

An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes.

An inflection point does not have to be a stationary point, but if it is, then it would also be a saddle point.

For a sufficiently differentiable function, a point is a saddle point if the smallest non-zero derivative is greater than $1$ and of odd order (extremum test).

For a twice differentiable function, a point is an inflection point if the second derivative changes sign around the point. A difference here is that the first derivative can be non-zero.

For example, for the function $f(x)=x^3+x$, $0$ is an inflection point but not a saddle point.

I resort to pathological examples such as $f(x) = \begin{cases} x^2 \sin\left(\frac1x\right) & x\ne0 \\ 0 & x=0 \end{cases}$ for a saddle point that is not an inflection point, since for elementary functions, a saddle point is an inflection point.

• Can inflection point be relative Maxima or minima ? @Kenny Lau Sep 26, 2017 at 19:20
• Inflection points [...] are not local maxima or local minima since the curve would be convex around a local minimum and concave around a local maximum. Sep 26, 2017 at 19:24
• I assumed the function to be twice differentiable. Sep 26, 2017 at 19:27
• Re: the final sentence: The point $(0,3)$ of the elementary function $f(x)=3$ is a saddle, but not inflection, point. Aug 17, 2022 at 9:34

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. - Wikipedia: Saddle point

• For functions with one variable a "stationary point that is not extremal" (extremal means either a minimum or a maximum) is a point where the function's first derivatives become zero and the second derivative changes sign.
• A saddle point is named after a horse saddle. So its visualizations are usually depicted as 3D with functions with two variables.
• Definition of a saddle point for functions with at least two variables is a point where the function has a zero derivative (or partial derivatives) and the function has different signs of curvature in different directions.

An example of a function with one variable that has an inflexion point which is also stationary (You may call it saddle point although the shape of it is a diagonal 2D projection of a 3D horse saddle.) is: $$f(x) = x^n, \quad n \in \mathbb{O}$$ For example, for the function $$f(x) = x^3$$, the point $$x = 0$$ is a saddle point because $$f’(x) = 0$$ but $$f(x)$$ is neither a maximum nor a minimum.

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May 19, 2023 at 6:26