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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731901.png" /> on a planar curve having the following properties: at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731902.png" /> the curve has a unique tangent, and within a small neighbourhood around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731903.png" /> the curve lies within one pair of vertical angles formed by the tangent and the normal (Fig. a). normal tangent
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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
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p073190a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p073190a.gif" />
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Figure: p073190a
 
Figure: p073190a
  
Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731904.png" /> be defined in a certain neighbourhood around a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731905.png" /> and let it be continuous at that point. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731906.png" /> is called a point of inflection for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731907.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731908.png" /> is called a point of inflection on the graph of the function, i.e. the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p0731909.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319010.png" />  "inflects"  through the tangent to it at that point; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319011.png" /> the tangent lies under the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319012.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319013.png" /> it lies above that graph (or vice versa, Fig. b).
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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).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p073190b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p073190b.gif" />
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Figure: p073190b
 
Figure: p073190b
  
A necessary existence condition for a point of inflection is: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319014.png" /> is twice differentiable in some neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319015.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319016.png" /> is a point of inflection, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319017.png" />. A sufficient existence condition for a point of inflection is: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319018.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319019.png" /> times continuously differentiable in a certain neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319020.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319021.png" /> odd and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319022.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319026.png" /> has a point of inflection at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073190/p07319027.png" />.
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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====
 
====References====
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====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


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)

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)
How to Cite This Entry:
Point of inflection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Point_of_inflection&oldid=48212