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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.

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  • $\begingroup$ 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. $\endgroup$
    – Paul
    Sep 26, 2017 at 19:23
  • $\begingroup$ This site has a clear diagram: quora.com/… $\endgroup$
    – NoChance
    May 19, 2023 at 6:17

2 Answers 2

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Saddle Point:

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

Inflection Point:

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.

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  • $\begingroup$ Can inflection point be relative Maxima or minima ? @Kenny Lau $\endgroup$
    – Zephyr
    Sep 26, 2017 at 19:20
  • $\begingroup$ 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. $\endgroup$
    – Kenny Lau
    Sep 26, 2017 at 19:24
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    $\begingroup$ I assumed the function to be twice differentiable. $\endgroup$
    – Kenny Lau
    Sep 26, 2017 at 19:27
  • $\begingroup$ Re: the final sentence: The point $(0,3)$ of the elementary function $f(x)=3$ is a saddle, but not inflection, point. $\endgroup$
    – ryang
    Aug 17, 2022 at 9:34
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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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  • $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    May 19, 2023 at 6:26

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