Point of inflection

A point $M$ on a planar curve having the following properties: at $M$ the curve has a unique tangent, and within a small neighbourhood around $M$ the curve lies within one pair of vertical angles formed by the tangent and the normal (Fig. a). normal tangent

Figure: p073190a

Let a function $f$ be defined in a certain neighbourhood around a point $x _ {0}$ and let it be continuous at that point. The point $x _ {0}$ is called a point of inflection for $f$ if it is simultaneously the end of a range of strict convexity upwards and the end of a range of strict convexity downwards. In that case the point $( x _ {0} , f( x _ {0} ))$ is called a point of inflection on the graph of the function, i.e. the graph of $f$ at $( x _ {0} , f( x _ {0} ))$" inflects" through the tangent to it at that point; for $x < x _ {0}$ the tangent lies under the graph of $f$, while for $x > x _ {0}$ it lies above that graph (or vice versa, Fig. b).

Figure: p073190b

A necessary existence condition for a point of inflection is: If $f$ is twice differentiable in some neighbourhood of a point $x _ {0}$, and if $x _ {0}$ is a point of inflection, then $f ^ { \prime\prime } ( x _ {0} ) = 0$. A sufficient existence condition for a point of inflection is: If $f$ is $k$ times continuously differentiable in a certain neighbourhood of a point $x$, with $k$ odd and $k \geq 3$, while $f ^ { ( n) } ( x _ {0} ) = 0$ for $n = 2 \dots k- 1$, and $f ^ { ( k) } ( x _ {0} ) \neq 0$, then $f$ has a point of inflection at $x _ {0}$.

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 2 , MIR (1982) (Translated from Russian)
[2]	L.D. Kudryavtsev, "A course in mathematical analysis" , 1 , Moscow (1981) (In Russian)

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959)