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 , they have found saddle points of a single variable function.
I am confused between both of these.
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$\begingroup$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.
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