482:
941:
133:
43:
2196:
1618:
1201:
2872:
1835:
740:
2603:
1959:
930:
2680:
1103:
3100:
Hughes-Hallett, Deborah; Lock, Patti Frazer; Gleason, Andrew M.; Flath, Daniel E.; Gordon, Sheldon P.; Lomen, David O.; Lovelock, David; McCallum, William G.; Osgood, Brad G. (2017-12-11).
2749:
1881:
2997:
There are two standard ways for using this fact. Firstly, if one knows, for example by physical measurement, the values of a function and its derivative at some sampling points, one can
2967:
2039:
1529:
876:
270:
1712:
1994:
1428:
762:
inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum. If
1289:
404:
2178:, its graph is symmetric with respect to the inflection point, and invariant under a rotation of a half turn around the inflection point. As these properties are invariant by
641:
2383:
3101:
1645:
2172:
2119:
2079:
1059:
987:
307:
to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its
1537:
1098:
447:. The graph of a cubic function is symmetric with respect to its inflection point; that is, it is invariant under a rotation of a half turn around this point.
2186:
The graph of a cubic function is symmetric with respect to its inflection point, and is invariant under a rotation of a half turn around the inflection point.
2760:
1730:
3024:
cubic. For having a uniquely defined interpolation, two more constraints must be added, such as the values of the derivatives at the endpoints, or a zero
3046:
53:
2990:
Given the values of a function and its derivative at two points, there is exactly one cubic function that has the same four values, which is called a
663:
2504:
1009:
Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always
3210:
3111:
3056:
3017:
3002:
2313:(in red) are the intersections of the (dotted) tangent lines to the graph at these points with the graph itself. They are collinear too.
3390:
1888:
884:
111:
68:
83:
3446:
3171:
878:
is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at
2614:
3385:
3369:
528:
440:
151:
3405:
3178:
3166:
90:
2691:
3250:
1840:
425:
3203:
2895:
97:
3395:
2003:
1218:
1077:
1467:
196:
1653:
3334:
3130:
1967:
1367:
1320:. After this change of variable, the new graph is the mirror image of the previous one, with respect of the
1224:
535:, that is the points where the slope of the function is zero. Thus the critical points of a cubic function
339:
79:
3364:
3359:
3354:
3232:
190:
3344:
3324:
2973:
2179:
1211:
1010:
991:
481:
452:
3410:
3329:
3196:
3161:
2991:
2985:
589:
312:
2334:
823:
3223:
3013:
999:
459:
432:
273:
1623:
3267:
3262:
3118:
A point at which the graph of the function f changes concavity is called an inflection point of f
2132:
2084:
2044:
1301:
1019:
947:
486:
418:
410:
140:
3400:
940:
3441:
3245:
3107:
3052:
818:
814:
654:
3420:
3298:
3291:
3286:
3182:
2318:
1613:{\displaystyle \textstyle x_{1}={\frac {x_{2}}{\sqrt {a}}},y_{1}={\frac {y_{2}}{\sqrt {a}}}}
774:
532:
502:
498:
494:
490:
436:
104:
1196:{\displaystyle {\begin{aligned}y&=x^{3}+x\\y&=x^{3}\\y&=x^{3}-x.\end{aligned}}}
3308:
3303:
3255:
3240:
1089:
1073:
2976:
that transforms collinear points into collinear points. This proves the claimed result.
3279:
3188:
3136:
Trilinear
Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions
2867:{\displaystyle (-2\alpha ,-8\alpha ^{3}-2p\alpha )=(-2\alpha ,-8f(\alpha )+6p\alpha ).}
471:
330:
304:
31:
1830:{\displaystyle \textstyle x_{2}=x_{3}{\sqrt {|p|}},\quad y_{2}=y_{3}{\sqrt {|p|^{3}}}}
3435:
3134:
3076:
300:
3415:
2325:
2175:
1081:
296:
428:); all odd-degree polynomials with real coefficients have at least one real root.
788:, then there are no (real) critical points. In the two latter cases, that is, if
1221:
can be built in the following way, when starting from a general cubic function
1065:
1003:
182:
137:
42:
17:
3219:
2195:
1092:
can transform the graph into the graph of one among the three cubic functions
581:
520:
414:
3349:
3025:
3021:
3006:
2321:
intercept the cubic again at collinear points. This can be seen as follows.
1069:
799:
735:{\displaystyle x_{\text{critical}}={\frac {-b\pm {\sqrt {b^{2}-3ac}}}{3a}}.}
444:
424:
A cubic function with real coefficients has either one or three real roots (
132:
2889:
of the graph to the other point where the tangent intercepts the graph is
2598:{\displaystyle x^{3}+px=\alpha ^{3}+p\alpha +(x-\alpha )(3\alpha ^{2}+p),}
519:(solid black curve) and its first (dashed red) and second (dotted orange)
303:
that maps real numbers to real numbers or as a complex function that maps
3339:
308:
3063:
Thus a cubic equation has either three real roots... or one real root...
443:, a local minimum and a local maximum. Otherwise, a cubic function is
60:
1620:
corresponds to a uniform scaling, and give, after multiplication by
2194:
2174:
the inflection point is thus the origin. As such a function is an
1207:
939:
813:
The inflection point of a function is where that function changes
448:
131:
1717:
which is the simplest form that can be obtained by a similarity.
3192:
1206:
This means that there are only three graphs of cubic functions
2459:
So, the intersection point between this line and the graph of
36:
2317:
The tangent lines to the graph of a cubic function at three
1954:{\displaystyle y_{3}=x_{3}^{3}+x_{3}\operatorname {sgn}(p),}
455:, there are only three possible graphs for cubic functions.
3045:
Bostock, Linda; Chandler, Suzanne; Chandler, F. S. (1979).
2041:
the latter form of the function applies to all cases (with
1006:, though many cubic curves are not graphs of functions.
653:-values of the critical points and are given, using the
3012:
If the value of a function is known at several points,
925:{\displaystyle x_{\text{inflection}}=-{\frac {b}{3a}}.}
64:
1844:
1734:
1541:
2898:
2763:
2694:
2617:
2507:
2337:
2135:
2087:
2047:
2006:
1970:
1891:
1843:
1733:
1656:
1626:
1540:
1470:
1370:
1227:
1101:
1022:
950:
887:
826:
773:, then there is only one critical point, which is an
666:
592:
342:
199:
3378:
3317:
3230:
2675:{\displaystyle x^{3}-3\alpha ^{2}x+2\alpha ^{3}=0,}
2392:is a real number, then the tangent to the graph of
2226:(in blue) are collinear and belong to the graph of
1064:This similarity can be built as the composition of
143:(where the curve crosses the horizontal axis—where
2961:
2866:
2743:
2674:
2597:
2377:
2328:, one may suppose that the function has the form
2166:
2113:
2073:
2033:
1988:
1953:
1875:
1829:
1706:
1639:
1612:
1523:
1433:This corresponds to a translation parallel to the
1422:
1283:
1195:
1053:
981:
924:
870:
734:
635:
398:
264:
3179:History of quadratic, cubic and quartic equations
3139:, Cambridge: Deighton, Bell, and Co., p. 425
2182:, the following is true for all cubic functions.
1457:corresponds to a translation with respect to the
1996:has the value 1 or –1, depending on the sign of
798:is nonpositive, the cubic function is strictly
3241:Zero polynomial (degree undefined or −1 or −∞)
2744:{\displaystyle (x-\alpha )^{2}(x+2\alpha )=0.}
3204:
1876:{\displaystyle \textstyle {\sqrt {|p|^{3}}},}
8:
3016:consists in approximating the function by a
802:. See the figure for an example of the case
69:introducing citations to additional sources
2962:{\displaystyle (x,y)\mapsto (-2x,-8y+6px).}
3211:
3197:
3189:
3133:(1866), "Equations of the third degree",
2897:
2789:
2762:
2711:
2693:
2657:
2638:
2622:
2616:
2577:
2534:
2512:
2506:
2357:
2336:
2146:
2134:
2105:
2092:
2086:
2065:
2052:
2046:
2034:{\displaystyle \operatorname {sgn}(0)=0,}
2005:
1969:
1927:
1914:
1909:
1896:
1890:
1861:
1856:
1847:
1845:
1842:
1818:
1813:
1804:
1802:
1796:
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1512:
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1369:
1257:
1241:
1226:
1174:
1150:
1120:
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1021:
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904:
892:
886:
825:
698:
692:
680:
671:
665:
603:
591:
366:
350:
341:
238:
222:
198:
2754:So, the tangent intercepts the cubic at
1524:{\displaystyle y_{1}=ax_{1}^{3}+px_{1}.}
1461:-axis, and gives a function of the form
480:
435:of a cubic function always has a single
265:{\displaystyle f(x)=ax^{3}+bx^{2}+cx+d,}
59:Relevant discussion may be found on the
3037:
1707:{\displaystyle y_{2}=x_{2}^{3}+px_{2},}
1013:to the graph of a function of the form
649:The solutions of this equation are the
3106:. John Wiley & Sons. p. 181.
2324:As this property is invariant under a
1989:{\displaystyle \operatorname {sgn}(p)}
1423:{\displaystyle y=ax_{1}^{3}+px_{1}+q.}
817:. An inflection point occurs when the
299:, and the function is considered as a
2463:can be obtained solving the equation
1284:{\displaystyle y=ax^{3}+bx^{2}+cx+d.}
399:{\displaystyle ax^{3}+bx^{2}+cx+d=0,}
7:
3018:continuously differentiable function
3003:continuously differentiable function
1068:parallel to the coordinates axes, a
458:Cubic functions are fundamental for
276:of degree three. In many texts, the
2877:So, the function that maps a point
990:The graph of any cubic function is
315:is restricted to the real numbers.
25:
2129:For a cubic function of the form
136:Graph of a cubic function with 3
1361:provides a function of the form
52:relies largely or entirely on a
41:
27:Polynomial function of degree 3
3051:. Nelson Thornes. p. 462.
1778:
646:of the cubic function is zero.
636:{\displaystyle 3ax^{2}+2bx+c=0}
2953:
2917:
2914:
2911:
2899:
2858:
2843:
2837:
2813:
2807:
2764:
2732:
2717:
2708:
2695:
2589:
2567:
2564:
2552:
2378:{\displaystyle f(x)=x^{3}+px.}
2347:
2341:
2019:
2013:
1983:
1977:
1945:
1939:
1857:
1848:
1814:
1805:
1769:
1761:
871:{\displaystyle f''(x)=6ax+2b,}
841:
835:
477:Critical and inflection points
209:
203:
1:
3401:Horner's method of evaluation
1327:Then, the change of variable
472:Cubic equation § History
1640:{\displaystyle {\sqrt {a}},}
944:Cubic functions of the form
531:of a cubic function are its
3406:Polynomial identity testing
3167:Encyclopedia of Mathematics
2167:{\displaystyle y=x^{3}+px,}
2114:{\displaystyle y_{2}=y_{3}}
2074:{\displaystyle x_{2}=x_{3}}
1054:{\displaystyle y=x^{3}+px.}
982:{\displaystyle y=x^{3}+cx.}
745:The sign of the expression
409:whose solutions are called
3463:
2983:
1727:, the non-uniform scaling
469:
150:). The case shown has two
29:
1837:gives, after division by
1219:geometric transformations
1002:of a cubic function is a
426:which may not be distinct
417:of a cubic function is a
3131:Whitworth, William Allen
1647:a function of the form
30:Not to be confused with
3391:Greatest common divisor
2608:which can be rewritten
1534:The change of variable
1440:The change of variable
154:. Here the function is
3263:Quadratic function (2)
2963:
2868:
2745:
2676:
2599:
2379:
2314:
2168:
2115:
2075:
2035:
1990:
1955:
1877:
1831:
1708:
1641:
1614:
1525:
1424:
1285:
1197:
1084:) with respect to the
1055:
995:
983:
926:
872:
736:
637:
524:
400:
266:
178:
3246:Constant function (0)
3081:mathworld.wolfram.com
2974:affine transformation
2964:
2869:
2746:
2677:
2600:
2380:
2198:
2169:
2116:
2076:
2036:
1991:
1956:
1878:
1832:
1709:
1642:
1615:
1526:
1425:
1286:
1212:affine transformation
1198:
1056:
984:
943:
927:
873:
737:
638:
484:
453:affine transformation
413:of the function. The
401:
267:
135:
3447:Polynomial functions
3379:Tools and algorithms
3299:Quintic function (5)
3287:Quartic function (4)
3224:polynomial functions
3001:the function with a
2992:cubic Hermite spline
2986:Spline interpolation
2896:
2761:
2692:
2615:
2505:
2335:
2133:
2085:
2045:
2004:
1968:
1889:
1841:
1731:
1654:
1624:
1538:
1468:
1368:
1225:
1099:
1076:), and, possibly, a
1020:
948:
885:
824:
664:
590:
340:
197:
65:improve this article
3309:Septic equation (7)
3304:Sextic equation (6)
3251:Linear function (1)
3075:Weisstein, Eric W.
3014:cubic interpolation
2980:Cubic interpolation
1919:
1684:
1501:
1394:
1090:non-uniform scaling
574:occur at values of
460:cubic interpolation
295:are supposed to be
274:polynomial function
3275:Cubic function (3)
3268:Quadratic equation
3077:"Stationary Point"
3048:Pure Mathematics 2
3028:at the endpoints.
2959:
2864:
2741:
2685:and factorized as
2672:
2595:
2375:
2315:
2164:
2111:
2071:
2031:
1986:
1951:
1905:
1873:
1872:
1827:
1826:
1704:
1670:
1637:
1610:
1609:
1521:
1487:
1420:
1380:
1302:change of variable
1281:
1193:
1191:
1051:
996:
979:
922:
868:
732:
633:
525:
439:. It may have two
419:quadratic function
396:
262:
179:
3429:
3428:
3370:Quasi-homogeneous
3162:"Cardano formula"
3113:978-1-119-27556-5
3058:978-0-85950-097-5
2000:. If one defines
1867:
1824:
1773:
1632:
1607:
1606:
1572:
1571:
1313:allows supposing
1088:-axis. A further
917:
895:
819:second derivative
727:
716:
674:
655:quadratic formula
533:stationary points
491:stationary points
130:
129:
115:
16:(Redirected from
3454:
3292:Quartic equation
3213:
3206:
3199:
3190:
3183:MacTutor archive
3175:
3148:
3147:
3146:
3144:
3127:
3121:
3120:
3103:Applied Calculus
3097:
3091:
3090:
3088:
3087:
3072:
3066:
3065:
3042:
3009:cubic function.
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2462:
2454:
2411:
2395:
2391:
2384:
2382:
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2376:
2362:
2361:
2319:collinear points
2312:
2303:
2294:
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2284:
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2256:
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2240:
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2118:
2117:
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2109:
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2096:
2080:
2078:
2077:
2072:
2070:
2069:
2057:
2056:
2040:
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2037:
2032:
1999:
1995:
1993:
1992:
1987:
1960:
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1952:
1932:
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1913:
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1900:
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1866:
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1836:
1834:
1833:
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1788:
1787:
1774:
1772:
1764:
1759:
1757:
1756:
1744:
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1713:
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1500:
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1460:
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1429:
1427:
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1421:
1410:
1409:
1393:
1388:
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1323:
1319:
1312:
1299:
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1287:
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1199:
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1192:
1179:
1178:
1155:
1154:
1125:
1124:
1087:
1060:
1058:
1057:
1052:
1038:
1037:
994:to such a curve.
988:
986:
985:
980:
966:
965:
931:
929:
928:
923:
918:
916:
905:
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869:
834:
809:
797:
787:
775:inflection point
772:
761:
752:
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607:
579:
569:
540:
518:
503:cubic polynomial
495:inflection point
437:inflection point
405:
403:
402:
397:
371:
370:
355:
354:
328:
311:, even when the
294:
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271:
269:
268:
263:
243:
242:
227:
226:
176:
149:
125:
122:
116:
114:
80:"Cubic function"
73:
45:
37:
21:
18:Cubic polynomial
3462:
3461:
3457:
3456:
3455:
3453:
3452:
3451:
3432:
3431:
3430:
3425:
3374:
3313:
3256:Linear equation
3226:
3217:
3160:
3157:
3152:
3151:
3142:
3140:
3129:
3128:
3124:
3114:
3099:
3098:
3094:
3085:
3083:
3074:
3073:
3069:
3059:
3044:
3043:
3039:
3034:
2988:
2982:
2894:
2893:
2878:
2785:
2759:
2758:
2707:
2690:
2689:
2653:
2634:
2618:
2613:
2612:
2573:
2530:
2508:
2503:
2502:
2464:
2460:
2416:
2397:
2393:
2389:
2353:
2333:
2332:
2311:
2305:
2302:
2296:
2293:
2287:
2279:
2276:
2273:
2272:
2270:
2260:
2257:
2254:
2253:
2251:
2241:
2238:
2235:
2234:
2232:
2227:
2224:
2218:
2215:
2209:
2206:
2200:
2193:
2142:
2131:
2130:
2127:
2101:
2088:
2083:
2082:
2061:
2048:
2043:
2042:
2002:
2001:
1997:
1966:
1965:
1923:
1892:
1887:
1886:
1855:
1839:
1838:
1812:
1792:
1779:
1748:
1735:
1729:
1728:
1721:
1691:
1657:
1652:
1651:
1622:
1621:
1592:
1577:
1557:
1542:
1536:
1535:
1508:
1471:
1466:
1465:
1458:
1451:
1441:
1434:
1401:
1366:
1365:
1351:
1348:
1343:
1342:
1340:
1338:
1328:
1321:
1314:
1304:
1294:
1253:
1237:
1223:
1222:
1190:
1189:
1170:
1163:
1157:
1156:
1146:
1139:
1133:
1132:
1116:
1109:
1097:
1096:
1085:
1074:uniform scaling
1029:
1018:
1017:
989:
957:
946:
945:
938:
909:
888:
883:
882:
827:
822:
821:
807:
803:
789:
778:
763:
753:
750:
746:
719:
694:
682:
667:
662:
661:
650:
599:
588:
587:
575:
545:
536:
529:critical points
505:
479:
474:
468:
441:critical points
362:
346:
338:
337:
319:
305:complex numbers
292:
288:
284:
280:
234:
218:
195:
194:
155:
152:critical points
144:
126:
120:
117:
74:
72:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
3460:
3458:
3450:
3449:
3444:
3434:
3433:
3427:
3426:
3424:
3423:
3418:
3413:
3408:
3403:
3398:
3393:
3388:
3382:
3380:
3376:
3375:
3373:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3321:
3319:
3315:
3314:
3312:
3311:
3306:
3301:
3296:
3295:
3294:
3284:
3283:
3282:
3280:Cubic equation
3272:
3271:
3270:
3260:
3259:
3258:
3248:
3243:
3237:
3235:
3228:
3227:
3218:
3216:
3215:
3208:
3201:
3193:
3187:
3186:
3176:
3156:
3155:External links
3153:
3150:
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3122:
3112:
3092:
3067:
3057:
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3035:
3033:
3030:
2984:Main article:
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2349:
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2343:
2340:
2309:
2300:
2291:
2222:
2213:
2204:
2192:
2191:Collinearities
2189:
2163:
2160:
2157:
2154:
2149:
2145:
2141:
2138:
2126:
2123:
2108:
2104:
2100:
2095:
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2068:
2064:
2060:
2055:
2051:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
1985:
1982:
1979:
1976:
1973:
1962:
1961:
1950:
1947:
1944:
1941:
1938:
1935:
1930:
1926:
1922:
1917:
1912:
1908:
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1268:
1265:
1260:
1256:
1252:
1249:
1244:
1240:
1236:
1233:
1230:
1204:
1203:
1188:
1185:
1182:
1177:
1173:
1169:
1166:
1164:
1162:
1159:
1158:
1153:
1149:
1145:
1142:
1140:
1138:
1135:
1134:
1131:
1128:
1123:
1119:
1115:
1112:
1110:
1108:
1105:
1104:
1062:
1061:
1050:
1047:
1044:
1041:
1036:
1032:
1028:
1025:
978:
975:
972:
969:
964:
960:
956:
953:
937:
936:Classification
934:
933:
932:
921:
915:
912:
908:
903:
900:
891:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
833:
830:
805:
748:
743:
742:
731:
725:
722:
715:
712:
709:
706:
701:
697:
691:
688:
685:
679:
670:
644:
643:
632:
629:
626:
623:
620:
617:
614:
611:
606:
602:
598:
595:
580:such that the
572:
571:
478:
475:
470:Main article:
467:
464:
407:
406:
395:
392:
389:
386:
383:
380:
377:
374:
369:
365:
361:
358:
353:
349:
345:
331:cubic equation
261:
258:
255:
252:
249:
246:
241:
237:
233:
230:
225:
221:
217:
214:
211:
208:
205:
202:
187:cubic function
128:
127:
121:September 2019
63:. Please help
49:
47:
40:
32:Cubic equation
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3459:
3448:
3445:
3443:
3440:
3439:
3437:
3422:
3421:Gröbner basis
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3386:Factorization
3384:
3383:
3381:
3377:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3322:
3320:
3318:By properties
3316:
3310:
3307:
3305:
3302:
3300:
3297:
3293:
3290:
3289:
3288:
3285:
3281:
3278:
3277:
3276:
3273:
3269:
3266:
3265:
3264:
3261:
3257:
3254:
3253:
3252:
3249:
3247:
3244:
3242:
3239:
3238:
3236:
3234:
3229:
3225:
3221:
3214:
3209:
3207:
3202:
3200:
3195:
3194:
3191:
3184:
3180:
3177:
3173:
3169:
3168:
3163:
3159:
3158:
3154:
3138:
3137:
3132:
3126:
3123:
3119:
3115:
3109:
3105:
3104:
3096:
3093:
3082:
3078:
3071:
3068:
3064:
3060:
3054:
3050:
3049:
3041:
3038:
3031:
3029:
3027:
3023:
3019:
3015:
3010:
3008:
3005:, which is a
3004:
3000:
2995:
2993:
2987:
2979:
2977:
2975:
2956:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2908:
2905:
2902:
2892:
2891:
2890:
2886:
2882:
2861:
2855:
2852:
2849:
2846:
2840:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2810:
2804:
2801:
2798:
2795:
2790:
2786:
2782:
2779:
2776:
2773:
2770:
2767:
2757:
2756:
2755:
2738:
2735:
2729:
2726:
2723:
2720:
2712:
2704:
2701:
2698:
2688:
2687:
2686:
2669:
2666:
2663:
2658:
2654:
2650:
2647:
2644:
2639:
2635:
2631:
2628:
2623:
2619:
2611:
2610:
2609:
2592:
2586:
2583:
2578:
2574:
2570:
2561:
2558:
2555:
2549:
2546:
2543:
2540:
2535:
2531:
2527:
2524:
2521:
2518:
2513:
2509:
2501:
2500:
2499:
2495:
2491:
2487:
2483:
2479:
2475:
2471:
2467:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2424:
2420:
2415:
2414:
2413:
2409:
2405:
2401:
2396:at the point
2372:
2369:
2366:
2363:
2358:
2354:
2350:
2344:
2338:
2331:
2330:
2329:
2327:
2322:
2320:
2308:
2299:
2290:
2286:. The points
2268:
2249:
2230:
2221:
2212:
2203:
2197:
2190:
2188:
2187:
2183:
2181:
2177:
2161:
2158:
2155:
2152:
2147:
2143:
2139:
2136:
2124:
2122:
2106:
2102:
2098:
2093:
2089:
2066:
2062:
2058:
2053:
2049:
2028:
2025:
2022:
2016:
2010:
2007:
1980:
1974:
1971:
1948:
1942:
1936:
1933:
1928:
1924:
1920:
1915:
1910:
1906:
1902:
1897:
1893:
1885:
1884:
1883:
1869:
1862:
1852:
1819:
1809:
1797:
1793:
1789:
1784:
1780:
1775:
1765:
1753:
1749:
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1736:
1724:
1718:
1701:
1696:
1692:
1688:
1685:
1680:
1675:
1671:
1667:
1662:
1658:
1650:
1649:
1648:
1634:
1629:
1603:
1597:
1593:
1587:
1582:
1578:
1574:
1568:
1562:
1558:
1552:
1547:
1543:
1518:
1513:
1509:
1505:
1502:
1497:
1492:
1488:
1484:
1481:
1476:
1472:
1464:
1463:
1462:
1455:
1448:
1444:
1438:
1417:
1414:
1411:
1406:
1402:
1398:
1395:
1390:
1385:
1381:
1377:
1374:
1371:
1364:
1363:
1362:
1355:
1346:
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1331:
1325:
1317:
1311:
1307:
1303:
1297:
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1278:
1275:
1272:
1269:
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1263:
1258:
1254:
1250:
1247:
1242:
1238:
1234:
1231:
1228:
1220:
1215:
1213:
1209:
1186:
1183:
1180:
1175:
1171:
1167:
1165:
1160:
1151:
1147:
1143:
1141:
1136:
1129:
1126:
1121:
1117:
1113:
1111:
1106:
1095:
1094:
1093:
1091:
1083:
1079:
1075:
1071:
1067:
1048:
1045:
1042:
1039:
1034:
1030:
1026:
1023:
1016:
1015:
1014:
1012:
1007:
1005:
1001:
993:
976:
973:
970:
967:
962:
958:
954:
951:
942:
935:
919:
913:
910:
906:
901:
898:
889:
881:
880:
879:
865:
862:
859:
856:
853:
850:
847:
844:
838:
831:
828:
820:
816:
811:
801:
796:
792:
785:
781:
776:
770:
766:
760:
756:
729:
723:
720:
713:
710:
707:
704:
699:
695:
689:
686:
683:
677:
668:
660:
659:
658:
656:
647:
630:
627:
624:
621:
618:
615:
612:
609:
604:
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586:
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584:
583:
578:
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534:
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516:
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456:
454:
450:
446:
442:
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434:
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420:
416:
412:
393:
390:
387:
384:
381:
378:
375:
372:
367:
363:
359:
356:
351:
347:
343:
336:
335:
334:
332:
326:
322:
316:
314:
310:
306:
302:
301:real function
298:
279:
275:
259:
256:
253:
250:
247:
244:
239:
235:
231:
228:
223:
219:
215:
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192:
188:
184:
174:
170:
166:
162:
158:
153:
147:
142:
139:
134:
124:
113:
110:
106:
103:
99:
96:
92:
89:
85:
82: –
81:
77:
76:Find sources:
70:
66:
62:
56:
55:
54:single source
50:This article
48:
44:
39:
38:
33:
19:
3416:Discriminant
3335:Multivariate
3274:
3165:
3141:, retrieved
3135:
3125:
3117:
3102:
3095:
3084:. Retrieved
3080:
3070:
3062:
3047:
3040:
3011:
2998:
2996:
2989:
2971:
2884:
2880:
2876:
2753:
2684:
2607:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2465:
2458:
2450:
2446:
2442:
2438:
2434:
2430:
2426:
2422:
2418:
2412:is the line
2407:
2403:
2399:
2387:
2326:rigid motion
2323:
2316:
2306:
2297:
2288:
2266:
2247:
2228:
2219:
2210:
2201:
2185:
2184:
2176:odd function
2128:
1963:
1722:
1719:
1716:
1533:
1453:
1446:
1442:
1439:
1432:
1353:
1344:
1333:
1329:
1326:
1315:
1309:
1305:
1295:
1293:Firstly, if
1292:
1216:
1205:
1082:mirror image
1066:translations
1063:
1008:
997:
812:
794:
790:
783:
779:
768:
764:
758:
754:
744:
648:
645:
576:
573:
566:
562:
558:
554:
550:
546:
541:defined by
537:
526:
514:
510:
506:
457:
430:
423:
408:
333:of the form
324:
320:
317:
297:real numbers
278:coefficients
277:
193:of the form
186:
180:
172:
168:
164:
160:
156:
145:
118:
108:
101:
94:
87:
75:
51:
3365:Homogeneous
3360:Square-free
3355:Irreducible
3220:Polynomials
3020:, which is
2999:interpolate
2972:This is an
2199:The points
1004:cubic curve
521:derivatives
329:produces a
272:that is, a
183:mathematics
3436:Categories
3325:Univariate
3086:2020-07-27
3032:References
2498:, that is
2445:)) :
2180:similarity
1217:The above
1078:reflection
894:inflection
582:derivative
415:derivative
91:newspapers
3411:Resultant
3350:Trinomial
3330:Bivariate
3172:EMS Press
3026:curvature
3022:piecewise
3007:piecewise
2933:−
2921:−
2915:↦
2856:α
2841:α
2829:−
2823:α
2817:−
2805:α
2796:−
2787:α
2780:−
2774:α
2768:−
2730:α
2705:α
2702:−
2655:α
2636:α
2629:−
2575:α
2562:α
2559:−
2547:α
2532:α
2492: ′(
2441: ′(
2011:
1975:
1937:
1720:Then, if
1181:−
1070:homothecy
902:−
815:concavity
800:monotonic
705:−
690:±
684:−
499:concavity
445:monotonic
61:talk page
3442:Calculus
3396:Division
3345:Binomial
3340:Monomial
3143:June 17,
2125:Symmetry
832:″
673:critical
318:Setting
309:codomain
191:function
3174:, 2001
2283:
2271:
2264:
2252:
2245:
2233:
1437:-axis.
1358:
1341:
1324:-axis.
1011:similar
992:similar
466:History
105:scholar
3233:degree
3110:
3055:
2304:, and
2217:, and
1964:where
1318:> 0
1300:, the
1298:< 0
808:> 0
786:< 0
657:, by
313:domain
291:, and
175:− 8)/4
107:
100:
93:
86:
78:
2480:) + (
2429:) + (
1208:up to
1000:graph
777:. If
501:of a
487:roots
449:Up to
433:graph
411:roots
327:) = 0
189:is a
163:) = (
141:roots
112:JSTOR
98:books
3222:and
3145:2016
3108:ISBN
3053:ISBN
2472:) =
2081:and
998:The
553:) =
527:The
497:and
485:The
431:The
185:, a
138:real
84:news
3231:By
3181:on
2388:If
2121:).
2008:sgn
1972:sgn
1934:sgn
1725:≠ 0
1308:→ –
1210:an
793:– 3
782:– 3
771:= 0
767:– 3
757:– 3
517:− 4
513:+ 9
509:− 6
451:an
181:In
171:− 6
167:+ 3
148:= 0
67:by
3438::
3170:,
3164:,
3116:.
3079:.
3061:.
2994:.
2883:,
2739:0.
2484:−
2449:∈
2433:−
2421:,
2417:{(
2410:))
2402:,
2295:,
2269:+
2250:−
2231:+
2208:,
1452:+
1445:=
1339:–
1332:=
1214:.
810:.
795:ac
784:ac
769:ac
759:ac
751:=
565:+
563:cx
561:+
559:bx
557:+
555:ax
493:,
489:,
462:.
421:.
287:,
283:,
3212:e
3205:t
3198:v
3185:.
3089:.
2957:.
2954:)
2951:x
2948:p
2945:6
2942:+
2939:y
2936:8
2930:,
2927:x
2924:2
2918:(
2912:)
2909:y
2906:,
2903:x
2900:(
2887:)
2885:y
2881:x
2879:(
2862:.
2859:)
2853:p
2850:6
2847:+
2844:)
2838:(
2835:f
2832:8
2826:,
2820:2
2814:(
2811:=
2808:)
2802:p
2799:2
2791:3
2783:8
2777:,
2771:2
2765:(
2736:=
2733:)
2727:2
2724:+
2721:x
2718:(
2713:2
2709:)
2699:x
2696:(
2670:,
2667:0
2664:=
2659:3
2651:2
2648:+
2645:x
2640:2
2632:3
2624:3
2620:x
2593:,
2590:)
2587:p
2584:+
2579:2
2571:3
2568:(
2565:)
2556:x
2553:(
2550:+
2544:p
2541:+
2536:3
2528:=
2525:x
2522:p
2519:+
2514:3
2510:x
2496:)
2494:α
2490:f
2488:)
2486:α
2482:x
2478:α
2476:(
2474:f
2470:x
2468:(
2466:f
2461:f
2455:.
2453:}
2451:R
2447:x
2443:α
2439:f
2437:)
2435:α
2431:x
2427:α
2425:(
2423:f
2419:x
2408:α
2406:(
2404:f
2400:α
2398:(
2394:f
2390:α
2373:.
2370:x
2367:p
2364:+
2359:3
2355:x
2351:=
2348:)
2345:x
2342:(
2339:f
2310:3
2307:T
2301:2
2298:T
2292:1
2289:T
2280:4
2277:/
2274:5
2267:x
2261:2
2258:/
2255:5
2248:x
2242:2
2239:/
2236:3
2229:x
2223:3
2220:P
2214:2
2211:P
2205:1
2202:P
2162:,
2159:x
2156:p
2153:+
2148:3
2144:x
2140:=
2137:y
2107:3
2103:y
2099:=
2094:2
2090:y
2067:3
2063:x
2059:=
2054:2
2050:x
2029:,
2026:0
2023:=
2020:)
2017:0
2014:(
1998:p
1984:)
1981:p
1978:(
1949:,
1946:)
1943:p
1940:(
1929:3
1925:x
1921:+
1916:3
1911:3
1907:x
1903:=
1898:3
1894:y
1870:,
1863:3
1858:|
1853:p
1849:|
1820:3
1815:|
1810:p
1806:|
1798:3
1794:y
1790:=
1785:2
1781:y
1776:,
1770:|
1766:p
1762:|
1754:3
1750:x
1746:=
1741:2
1737:x
1723:p
1702:,
1697:2
1693:x
1689:p
1686:+
1681:3
1676:2
1672:x
1668:=
1663:2
1659:y
1635:,
1630:a
1604:a
1598:2
1594:y
1588:=
1583:1
1579:y
1575:,
1569:a
1563:2
1559:x
1553:=
1548:1
1544:x
1519:.
1514:1
1510:x
1506:p
1503:+
1498:3
1493:1
1489:x
1485:a
1482:=
1477:1
1473:y
1459:y
1454:q
1450:1
1447:y
1443:y
1435:x
1418:.
1415:q
1412:+
1407:1
1403:x
1399:p
1396:+
1391:3
1386:1
1382:x
1378:a
1375:=
1372:y
1354:a
1352:3
1349:/
1345:b
1337:1
1334:x
1330:x
1322:y
1316:a
1310:x
1306:x
1296:a
1279:.
1276:d
1273:+
1270:x
1267:c
1264:+
1259:2
1255:x
1251:b
1248:+
1243:3
1239:x
1235:a
1232:=
1229:y
1187:.
1184:x
1176:3
1172:x
1168:=
1161:y
1152:3
1148:x
1144:=
1137:y
1130:x
1127:+
1122:3
1118:x
1114:=
1107:y
1086:y
1080:(
1072:(
1049:.
1046:x
1043:p
1040:+
1035:3
1031:x
1027:=
1024:y
977:.
974:x
971:c
968:+
963:3
959:x
955:=
952:y
920:.
914:a
911:3
907:b
899:=
890:x
866:,
863:b
860:2
857:+
854:x
851:a
848:6
845:=
842:)
839:x
836:(
829:f
806:0
804:Δ
791:b
780:b
765:b
755:b
749:0
747:Δ
730:.
724:a
721:3
714:c
711:a
708:3
700:2
696:b
687:b
678:=
669:x
651:x
631:0
628:=
625:c
622:+
619:x
616:b
613:2
610:+
605:2
601:x
597:a
594:3
577:x
570:,
567:d
551:x
549:(
547:f
538:f
523:.
515:x
511:x
507:x
394:,
391:0
388:=
385:d
382:+
379:x
376:c
373:+
368:2
364:x
360:b
357:+
352:3
348:x
344:a
325:x
323:(
321:f
293:d
289:c
285:b
281:a
260:,
257:d
254:+
251:x
248:c
245:+
240:2
236:x
232:b
229:+
224:3
220:x
216:a
213:=
210:)
207:x
204:(
201:f
177:.
173:x
169:x
165:x
161:x
159:(
157:f
146:y
123:)
119:(
109:·
102:·
95:·
88:·
71:.
57:.
34:.
20:)
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