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 '17 at 19:23

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 – Zephyr Sep 26 '17 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. – Kenny Lau Sep 26 '17 at 19:24
• Actually I am finding lot of contradicting things in Wolfram site. math.stackexchange.com/questions/80655/…. Here the answer says it's possible but Wolfram says it's not possible. – Zephyr Sep 26 '17 at 19:25
• I assumed the function to be twice differentiable. – Kenny Lau Sep 26 '17 at 19:27