36:
115:
840:
93:
588:
for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but
1047:
Some continuous functions have an inflection point even though the second derivative is never 0. For example, the cube root function is concave upward when x is negative, and concave downward when x is positive, but has no derivatives of any order at the origin.
434:
of the tangent line and the curve (at the point of tangency) is greater than 2. The main motivation of this different definition, is that otherwise the set of the inflection points of a curve would not be an
856:
has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well).
274:
and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a
1029:
998:
Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function
389:
is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A
455:
393:
is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.
290:
633:, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of
1112:
286:
57:
927:
1183:
79:
1269:
411:
For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the
1254:
1249:
1136:
1232:
1057:
1227:
50:
44:
1259:
1160:
827:
652:
454:
61:
1001:
313:
218:
1222:
448:
165:
161:
114:
1062:
585:
1067:
444:
431:
397:
197:
593:. However, in algebraic geometry, both inflection points and undulation points are usually called
1154:
440:
419:
405:
119:
261:
from a positive value (concave upward) to a negative value (concave downward) or vice versa as
1204:
1179:
1142:
1132:
1108:
412:
233:
894:
201:
193:
135:
131:
123:
107:
1264:
668:
660:
505:
423:
205:
1082:
1071:
923:
439:. In fact, the set of the inflection points of a plane algebraic curve are exactly its
371:
340:
1243:
1207:
1039:, but it has no points of inflection because 0 is not in the domain of the function.
436:
839:
931:
509:
285:
In algebraic geometry an inflection point is defined slightly more generally, as a
305:
Inflection points in differential geometry are the points of the curve where the
17:
189:
482:
333:
153:
512:), green where concave (below its tangent), and red at inflection points: 0,
297:
is defined as a point where the tangent meets the curve to order at least 4.
1212:
1146:
684:
A sufficient existence condition for a point of inflection in the case that
401:
306:
152:(solid black curve) and its first (dashed red) and second (dotted orange)
92:
363:
343:
1043:
Functions with inflection points whose second derivative does not vanish
699:
times continuously differentiable in a certain neighborhood of a point
379:
860:
Points of inflection can also be categorized according to whether
838:
453:
91:
1076:
958:
An example of a non-stationary point of inflection is the point
404:
changes from plus to minus or from minus to plus, i.e., changes
930:, a stationary point that is not a local extremum is called a
248:
can also be used to find an inflection point since a point of
29:
781:
Another more general sufficient existence condition requires
937:
An example of a stationary point of inflection is the point
267:
is continuous; an inflection point of the curve is where
374:, then an inflection point is a point on the graph of
200:, it is a point where the function changes from being
1107:(8 ed.). Boston: Cengage Learning. p. 281.
1004:
1131:. Baranenkov, G. S. Moscow: Mir Publishers. 1976 .
817:
to have opposite signs in the neighborhood of
500:, and its sign is thus the opposite of the sign of
106:with an inflection point at (0,0), which is also a
1023:
354:has an extremum). That is, in some neighborhood,
27:Point where the curvature of a curve changes sign
1079:, an architectural form with an inflection point
625:In the preceding assertions, it is assumed that
196:changes sign. In particular, in the case of the
1178:(4th ed.). Berlin: Springer. p. 231.
629:has some higher-order non-zero derivative at
8:
1070:formed by the nine inflection points of an
955:-axis, which cuts the graph at this point.
241:, exist and are continuous), the condition
922:A stationary point of inflection is not a
1085:, a local minimum or maximum of curvature
1011:
1003:
362:has a (local) minimum or maximum. If all
80:Learn how and when to remove this message
980:. The tangent at the origin is the line
529:A necessary but not sufficient condition
400:, a point is an inflection point if its
113:
43:This article includes a list of general
1095:
1024:{\displaystyle x\mapsto {\frac {1}{x}}}
597:. An example of an undulation point is
415:has an isolated zero and changes sign.
350:. (this is not the same as saying that
293:at least 3, and an undulation point or
1152:
990:, which cuts the graph at this point.
835:Categorization of points of inflection
504:. Tangent is blue where the curve is
289:where the tangent meets the curve to
7:
926:. More generally, in the context of
928:functions of several real variables
358:is the one and only point at which
917:non-stationary point of inflection
49:it lacks sufficient corresponding
25:
1129:Problems in mathematical analysis
208:(concave upward), or vice versa.
1174:Bronshtein; Semendyayev (2004).
34:
1008:
994:Functions with discontinuities
647:is the same on either side of
312:For example, the graph of the
1:
769:has a point of inflection at
422:, a non singular point of an
1058:Critical point (mathematics)
915:is not zero, the point is a
584:, but this condition is not
396:For a smooth curve given by
211:For the graph of a function
1228:Encyclopedia of Mathematics
673:falling point of inflection
566:is an inflection point for
537:, if its second derivative
387:falling point of inflection
316:has an inflection point at
1286:
828:Bronshtein and Semendyayev
665:rising point of inflection
391:rising point of inflection
255:must be passed to change
1035:and convex for positive
1031:is concave for negative
892:is zero, the point is a
1270:Curvature (mathematics)
1176:Handbook of Mathematics
1103:Stewart, James (2015).
314:differentiable function
219:differentiability class
64:more precise citations.
1159:: CS1 maint: others (
1025:
857:
520:
443:that are zeros of the
226:(its first derivative
204:(concave downward) to
157:
111:
1255:Differential geometry
1250:Differential calculus
1223:"Point of inflection"
1026:
951:. The tangent is the
842:
679:Sufficient conditions
457:
449:projective completion
166:differential geometry
162:differential calculus
117:
95:
1063:Ecological threshold
1002:
874:is zero or nonzero.
398:parametric equations
1068:Hesse configuration
830:2004, p. 231).
445:Hessian determinant
441:non-singular points
432:intersection number
430:if and only if the
382:crosses the curve.
332:if and only if its
276:point of undulation
198:graph of a function
174:point of inflection
1208:"Inflection Point"
1205:Weisstein, Eric W.
1021:
976:, for any nonzero
858:
659:. If this sign is
521:
420:algebraic geometry
309:changes its sign.
190:smooth plane curve
188:) is a point on a
158:
112:
1260:Analytic geometry
1114:978-1-285-74062-1
1019:
671:, the point is a
663:, the point is a
604:for the function
595:inflection points
413:second derivative
234:second derivative
124:stationary points
90:
89:
82:
18:Inflection points
16:(Redirected from
1277:
1236:
1218:
1217:
1190:
1189:
1171:
1165:
1164:
1158:
1150:
1125:
1119:
1118:
1100:
1038:
1034:
1030:
1028:
1027:
1022:
1020:
1012:
989:
979:
975:
962:on the graph of
961:
954:
950:
941:on the graph of
940:
914:
907:
895:stationary point
891:
884:
873:
866:
855:
825:
816:
798:
777:
768:
757:
743:
732:
718:
711:
707:
698:
694:
658:
650:
646:
639:
632:
628:
621:
607:
603:
591:undulation point
583:
569:
565:
556:
547:
519:
515:
503:
499:
480:
476:
472:
428:inflection point
402:signed curvature
377:
369:
361:
357:
353:
349:
338:
334:first derivative
331:
280:undulation point
273:
266:
260:
254:
247:
240:
231:
225:
216:
170:inflection point
151:
136:cubic polynomial
128:inflection point
108:stationary point
105:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
1285:
1284:
1280:
1279:
1278:
1276:
1275:
1274:
1240:
1239:
1221:
1203:
1202:
1199:
1194:
1193:
1186:
1173:
1172:
1168:
1151:
1139:
1127:
1126:
1122:
1115:
1102:
1101:
1097:
1092:
1054:
1045:
1036:
1032:
1000:
999:
996:
981:
977:
963:
959:
952:
942:
938:
905:
902:
882:
879:
864:
861:
843:
837:
824:
818:
810:
800:
792:
782:
776:
770:
759:
755:
745:
734:
730:
720:
713:
709:
706:
700:
696:
685:
681:
656:
648:
637:
634:
630:
626:
609:
605:
598:
581:
571:
567:
564:
558:
555:
549:
538:
533:For a function
531:
526:
517:
513:
508:(above its own
501:
486:
481:/4; the second
478:
474:
459:
424:algebraic curve
375:
367:
359:
355:
351:
347:
336:
317:
303:
268:
262:
256:
249:
242:
236:
227:
221:
212:
138:
97:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
1283:
1281:
1273:
1272:
1267:
1262:
1257:
1252:
1242:
1241:
1238:
1237:
1219:
1198:
1195:
1192:
1191:
1184:
1166:
1137:
1120:
1113:
1094:
1093:
1091:
1088:
1087:
1086:
1083:Vertex (curve)
1080:
1074:
1072:elliptic curve
1065:
1060:
1053:
1050:
1044:
1041:
1018:
1015:
1010:
1007:
995:
992:
924:local extremum
920:
919:
899:
836:
833:
832:
831:
822:
808:
790:
779:
774:
753:
728:
704:
680:
677:
579:
562:
553:
530:
527:
525:
522:
302:
299:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1282:
1271:
1268:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1247:
1245:
1234:
1230:
1229:
1224:
1220:
1215:
1214:
1209:
1206:
1201:
1200:
1196:
1187:
1185:3-540-43491-7
1181:
1177:
1170:
1167:
1162:
1156:
1148:
1144:
1140:
1134:
1130:
1124:
1121:
1116:
1110:
1106:
1099:
1096:
1089:
1084:
1081:
1078:
1075:
1073:
1069:
1066:
1064:
1061:
1059:
1056:
1055:
1051:
1049:
1042:
1040:
1016:
1013:
1005:
993:
991:
988:
984:
974:
970:
966:
956:
949:
945:
935:
933:
929:
925:
918:
912:
908:
900:
898:
897:of inflection
896:
889:
885:
877:
876:
875:
871:
867:
854:
850:
846:
841:
834:
829:
821:
814:
807:
803:
796:
789:
785:
780:
773:
766:
762:
752:
748:
741:
737:
727:
723:
716:
703:
692:
688:
683:
682:
678:
676:
674:
670:
666:
662:
654:
644:
640:
623:
620:
616:
612:
601:
596:
592:
587:
578:
574:
561:
552:
545:
541:
536:
528:
523:
511:
507:
497:
493:
489:
484:
470:
466:
462:
456:
452:
450:
446:
442:
438:
437:algebraic set
433:
429:
425:
421:
416:
414:
409:
407:
403:
399:
394:
392:
388:
383:
381:
378:at which the
373:
365:
345:
342:
335:
329:
325:
321:
315:
310:
308:
300:
298:
296:
292:
288:
287:regular point
283:
281:
277:
271:
265:
259:
252:
245:
239:
235:
230:
224:
220:
215:
209:
207:
203:
199:
195:
192:at which the
191:
187:
183:
179:
175:
171:
167:
163:
155:
149:
145:
141:
137:
133:
129:
125:
121:
116:
109:
104:
100:
94:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
1226:
1211:
1175:
1169:
1128:
1123:
1104:
1098:
1046:
997:
986:
982:
972:
968:
964:
957:
947:
943:
936:
932:saddle point
921:
916:
910:
903:
893:
887:
880:
869:
862:
859:
852:
848:
844:
819:
812:
805:
801:
794:
787:
783:
771:
764:
760:
750:
746:
739:
735:
725:
721:
714:
701:
690:
686:
672:
664:
653:neighborhood
642:
635:
624:
618:
614:
610:
599:
594:
590:
576:
572:
559:
550:
543:
539:
534:
532:
495:
491:
487:
468:
464:
460:
427:
417:
410:
395:
390:
386:
384:
327:
323:
319:
311:
304:
294:
284:
279:
275:
269:
263:
257:
250:
243:
237:
228:
222:
213:
210:
185:
181:
177:
173:
169:
159:
147:
143:
139:
127:
102:
98:
76:
67:
48:
719:, is that
667:; if it is
494:) = –4sin(2
154:derivatives
62:introducing
1244:Categories
1138:5030009434
1090:References
738:= 2, ...,
586:sufficient
548:exists at
524:Conditions
483:derivative
301:Definition
232:, and its
182:inflection
45:references
1233:EMS Press
1213:MathWorld
1155:cite book
1009:↦
608:given by
467:) = sin(2
307:curvature
295:hyperflex
194:curvature
186:inflexion
132:concavity
70:July 2013
1147:21598952
1105:Calculus
1052:See also
802:f″
784:f″
712:odd and
669:negative
661:positive
573:f″
540:f″
488:f″
458:Plot of
372:isolated
344:extremum
341:isolated
184:(rarely
96:Plot of
1235:, 2001
1197:Sources
758:. Then
570:, then
516:/2 and
510:tangent
477:/4 to 5
447:of its
380:tangent
364:extrema
339:has an
202:concave
58:improve
1265:Curves
1182:
1145:
1135:
1111:
960:(0, 0)
939:(0, 0)
506:convex
473:from −
426:is an
270:f'' =
244:f'' =
206:convex
47:, but
906:'
883:'
865:'
756:) ≠ 0
731:) = 0
708:with
651:in a
638:'
582:) = 0
291:order
251:f'' =
180:, or
168:, an
134:of a
120:roots
1180:ISBN
1161:link
1143:OCLC
1133:ISBN
1109:ISBN
1077:Ogee
799:and
744:and
733:for
617:) =
557:and
406:sign
370:are
178:flex
164:and
130:and
118:The
901:if
878:if
742:− 1
717:≥ 3
695:is
655:of
602:= 0
589:an
485:is
418:In
366:of
346:at
278:or
264:f''
258:f''
238:f''
217:of
160:In
150:− 4
146:+ 9
142:− 6
1246::
1231:,
1225:,
1210:.
1157:}}
1153:{{
1141:.
987:ax
985:=
973:ax
971:+
967:=
946:=
934:.
851:–
847:=
811:−
793:+
675:.
622:.
451:.
408:.
385:A
368:f'
360:f'
337:f'
330:))
322:,
282:.
229:f'
176:,
172:,
126:,
122:,
101:=
1216:.
1188:.
1163:)
1149:.
1117:.
1037:x
1033:x
1017:x
1014:1
1006:x
983:y
978:a
969:x
965:y
953:x
948:x
944:y
913:)
911:x
909:(
904:f
890:)
888:x
886:(
881:f
872:)
870:x
868:(
863:f
853:x
849:x
845:y
826:(
823:0
820:x
815:)
813:ε
809:0
806:x
804:(
797:)
795:ε
791:0
788:x
786:(
778:.
775:0
772:x
767:)
765:x
763:(
761:f
754:0
751:x
749:(
747:f
740:k
736:n
729:0
726:x
724:(
722:f
715:k
710:k
705:0
702:x
697:k
693:)
691:x
689:(
687:f
657:x
649:x
645:)
643:x
641:(
636:f
631:x
627:f
619:x
615:x
613:(
611:f
606:f
600:x
580:0
577:x
575:(
568:f
563:0
560:x
554:0
551:x
546:)
544:x
542:(
535:f
518:π
514:π
502:f
498:)
496:x
492:x
490:(
479:π
475:π
471:)
469:x
465:x
463:(
461:f
376:f
356:x
352:f
348:x
328:x
326:(
324:f
320:x
318:(
272:0
253:0
246:0
223:C
214:f
156:.
148:x
144:x
140:x
110:.
103:x
99:y
83:)
77:(
72:)
68:(
54:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
