Point of inflection
A point on a planar curve having the following properties: at
the curve has a unique tangent, and within a small neighbourhood around
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 be defined in a certain neighbourhood around a point
and let it be continuous at that point. The point
is called a point of inflection for
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
is called a point of inflection on the graph of the function, i.e. the graph of
at
"inflects" through the tangent to it at that point; for
the tangent lies under the graph of
, while for
it lies above that graph (or vice versa, Fig. b).
Figure: p073190b
A necessary existence condition for a point of inflection is: If is twice differentiable in some neighbourhood of a point
, and if
is a point of inflection, then
. A sufficient existence condition for a point of inflection is: If
is
times continuously differentiable in a certain neighbourhood of a point
, with
odd and
, while
for
, and
, then
has a point of inflection at
.
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) |
Comments
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) |
Point of inflection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Point_of_inflection&oldid=14389