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Der Graph ist eine ",[569,765,766],{},"Linkskurve"," (linksgekrümmt, ",[569,769,770],{},"konvex",").",[577,773,774,731,878,906,907,910,911,771],{},[580,775,777,810],{"className":776},[583],[580,778,780],{"className":779},[587],[589,781,782],{"xmlns":591},[593,783,784,807],{},[596,785,786,796,798,800,802,805],{},[599,787,788,790],{},[602,789,604],{},[596,791,792,794],{},[608,793,611],{"mathvariant":610},[608,795,611],{"mathvariant":610},[608,797,617],{"stretchy":616},[602,799,620],{},[608,801,623],{"stretchy":616},[608,803,804],{},"\u003C",[628,806,630],{},[632,808,809],{"encoding":634},"f''(x) \u003C 0",[580,811,813,869],{"className":812,"ariaHidden":640},[639],[580,814,816,819,851,854,857,860,863,866],{"className":815},[644],[580,817],{"className":818,"style":649},[648],[580,820,822,825],{"className":821},[653],[580,823,604],{"className":824,"style":658},[653,657],[580,826,828],{"className":827},[662],[580,829,831],{"className":830},[666],[580,832,834],{"className":833},[670],[580,835,837],{"className":836,"style":675},[674],[580,838,839,842],{"style":678},[580,840],{"className":841,"style":683},[682],[580,843,845],{"className":844},[687,688,689,690],[580,846,848],{"className":847},[653,690],[580,849,697],{"className":850},[653,690],[580,852,617],{"className":853},[701],[580,855,620],{"className":856},[653,657],[580,858,623],{"className":859},[708],[580,861],{"className":862,"style":713},[712],[580,864,804],{"className":865},[717],[580,867],{"className":868,"style":713},[712],[580,870,872,875],{"className":871},[644],[580,873],{"className":874,"style":727},[648],[580,876,630],{"className":877},[653],[580,879,881,894],{"className":880},[583],[580,882,884],{"className":883},[587],[589,885,886],{"xmlns":591},[593,887,888,892],{},[596,889,890],{},[602,891,746],{},[632,893,746],{"encoding":634},[580,895,897],{"className":896,"ariaHidden":640},[639],[580,898,900,903],{"className":899},[644],[580,901],{"className":902,"style":758},[648],[580,904,746],{"className":905,"style":762},[653,657],": Die Steigung nimmt ab. Der Graph ist eine ",[569,908,909],{},"Rechtskurve"," (rechtsgekrümmt, ",[569,912,913],{},"konkav",[565,915,916,917,946],{},"Anschaulich: Fährt man mit dem Fahrrad in Richtung wachsender ",[580,918,920,933],{"className":919},[583],[580,921,923],{"className":922},[587],[589,924,925],{"xmlns":591},[593,926,927,931],{},[596,928,929],{},[602,930,620],{},[632,932,620],{"encoding":634},[580,934,936],{"className":935,"ariaHidden":640},[639],[580,937,939,943],{"className":938},[644],[580,940],{"className":941,"style":942},[648],"height:0.4306em;",[580,944,620],{"className":945},[653,657],"-Werte über den Graphen, muss man bei einer Linkskurve nach links lenken.",[565,948,949,950,953,954,1058,1059,1158,1159,1187],{},"Das Kriterium ist ",[569,951,952],{},"hinreichend, aber nicht notwendig",". Einzelne Stellen mit ",[580,955,957,990],{"className":956},[583],[580,958,960],{"className":959},[587],[589,961,962],{"xmlns":591},[593,963,964,987],{},[596,965,966,976,978,980,982,985],{},[599,967,968,970],{},[602,969,604],{},[596,971,972,974],{},[608,973,611],{"mathvariant":610},[608,975,611],{"mathvariant":610},[608,977,617],{"stretchy":616},[602,979,620],{},[608,981,623],{"stretchy":616},[608,983,984],{},"=",[628,986,630],{},[632,988,989],{"encoding":634},"f''(x) = 0",[580,991,993,1049],{"className":992,"ariaHidden":640},[639],[580,994,996,999,1031,1034,1037,1040,1043,1046],{"className":995},[644],[580,997],{"className":998,"style":649},[648],[580,1000,1002,1005],{"className":1001},[653],[580,1003,604],{"className":1004,"style":658},[653,657],[580,1006,1008],{"className":1007},[662],[580,1009,1011],{"className":1010},[666],[580,1012,1014],{"className":1013},[670],[580,1015,1017],{"className":1016,"style":675},[674],[580,1018,1019,1022],{"style":678},[580,1020],{"className":1021,"style":683},[682],[580,1023,1025],{"className":1024},[687,688,689,690],[580,1026,1028],{"className":1027},[653,690],[580,1029,697],{"className":1030},[653,690],[580,1032,617],{"className":1033},[701],[580,1035,620],{"className":1036},[653,657],[580,1038,623],{"className":1039},[708],[580,1041],{"className":1042,"style":713},[712],[580,1044,984],{"className":1045},[717],[580,1047],{"className":1048,"style":713},[712],[580,1050,1052,1055],{"className":1051},[644],[580,1053],{"className":1054,"style":727},[648],[580,1056,630],{"className":1057},[653]," können in einer Linkskurve liegen, z.B. bei ",[580,1060,1062,1091],{"className":1061},[583],[580,1063,1065],{"className":1064},[587],[589,1066,1067],{"xmlns":591},[593,1068,1069,1088],{},[596,1070,1071,1073,1075,1077,1079,1081],{},[602,1072,604],{},[608,1074,617],{"stretchy":616},[602,1076,620],{},[608,1078,623],{"stretchy":616},[608,1080,984],{},[599,1082,1083,1085],{},[602,1084,620],{},[628,1086,1087],{},"4",[632,1089,1090],{"encoding":634},"f(x) = x^4",[580,1092,1094,1122],{"className":1093,"ariaHidden":640},[639],[580,1095,1097,1101,1104,1107,1110,1113,1116,1119],{"className":1096},[644],[580,1098],{"className":1099,"style":1100},[648],"height:1em;vertical-align:-0.25em;",[580,1102,604],{"className":1103,"style":658},[653,657],[580,1105,617],{"className":1106},[701],[580,1108,620],{"className":1109},[653,657],[580,1111,623],{"className":1112},[708],[580,1114],{"className":1115,"style":713},[712],[580,1117,984],{"className":1118},[717],[580,1120],{"className":1121,"style":713},[712],[580,1123,1125,1129],{"className":1124},[644],[580,1126],{"className":1127,"style":1128},[648],"height:0.8141em;",[580,1130,1132,1135],{"className":1131},[653],[580,1133,620],{"className":1134},[653,657],[580,1136,1138],{"className":1137},[662],[580,1139,1141],{"className":1140},[666],[580,1142,1144],{"className":1143},[670],[580,1145,1147],{"className":1146,"style":1128},[674],[580,1148,1149,1152],{"style":678},[580,1150],{"className":1151,"style":683},[682],[580,1153,1155],{"className":1154},[687,688,689,690],[580,1156,1087],{"className":1157},[653,690]," an der Stelle ",[580,1160,1162,1175],{"className":1161},[583],[580,1163,1165],{"className":1164},[587],[589,1166,1167],{"xmlns":591},[593,1168,1169,1173],{},[596,1170,1171],{},[628,1172,630],{},[632,1174,630],{"encoding":634},[580,1176,1178],{"className":1177,"ariaHidden":640},[639],[580,1179,1181,1184],{"className":1180},[644],[580,1182],{"className":1183,"style":727},[648],[580,1185,630],{"className":1186},[653],".",[565,1189,1190],{},"Eine Linkskurve liegt oberhalb jeder ihrer Tangenten und unterhalb jeder Sekante zwischen zwei ihrer Punkte, bei einer Rechtskurve ist es umgekehrt. Die Stellen, an denen die Krümmung wechselt, sind die Wendestellen.",[560,1192,1194],{"id":1193},"beispiele","Beispiele",[565,1196,1197,1325,1326,1435,1436,1488,1489,1591,1592,1707,1708,1488,1759,1591,1861,1966,1967,2018],{},[580,1198,1200,1236],{"className":1199},[583],[580,1201,1203],{"className":1202},[587],[589,1204,1205],{"xmlns":591},[593,1206,1207,1233],{},[596,1208,1209,1211,1213,1215,1217,1219,1226,1229,1231],{},[602,1210,604],{},[608,1212,617],{"stretchy":616},[602,1214,620],{},[608,1216,623],{"stretchy":616},[608,1218,984],{},[599,1220,1221,1223],{},[602,1222,620],{},[628,1224,1225],{},"3",[608,1227,1228],{},"−",[628,1230,1225],{},[602,1232,620],{},[632,1234,1235],{"encoding":634},"f(x) = x^3 - 3x",[580,1237,1239,1266,1313],{"className":1238,"ariaHidden":640},[639],[580,1240,1242,1245,1248,1251,1254,1257,1260,1263],{"className":1241},[644],[580,1243],{"className":1244,"style":1100},[648],[580,1246,604],{"className":1247,"style":658},[653,657],[580,1249,617],{"className":1250},[701],[580,1252,620],{"className":1253},[653,657],[580,1255,623],{"className":1256},[708],[580,1258],{"className":1259,"style":713},[712],[580,1261,984],{"className":1262},[717],[580,1264],{"className":1265,"style":713},[712],[580,1267,1269,1273,1302,1306,1310],{"className":1268},[644],[580,1270],{"className":1271,"style":1272},[648],"height:0.8974em;vertical-align:-0.0833em;",[580,1274,1276,1279],{"className":1275},[653],[580,1277,620],{"className":1278},[653,657],[580,1280,1282],{"className":1281},[662],[580,1283,1285],{"className":1284},[666],[580,1286,1288],{"className":1287},[670],[580,1289,1291],{"className":1290,"style":1128},[674],[580,1292,1293,1296],{"style":678},[580,1294],{"className":1295,"style":683},[682],[580,1297,1299],{"className":1298},[687,688,689,690],[580,1300,1225],{"className":1301},[653,690],[580,1303],{"className":1304,"style":1305},[712],"margin-right:0.2222em;",[580,1307,1228],{"className":1308},[1309],"mbin",[580,1311],{"className":1312,"style":1305},[712],[580,1314,1316,1319,1322],{"className":1315},[644],[580,1317],{"className":1318,"style":727},[648],[580,1320,1225],{"className":1321},[653],[580,1323,620],{"className":1324},[653,657]," mit ",[580,1327,1329,1364],{"className":1328},[583],[580,1330,1332],{"className":1331},[587],[589,1333,1334],{"xmlns":591},[593,1335,1336,1361],{},[596,1337,1338,1348,1350,1352,1354,1356,1359],{},[599,1339,1340,1342],{},[602,1341,604],{},[596,1343,1344,1346],{},[608,1345,611],{"mathvariant":610},[608,1347,611],{"mathvariant":610},[608,1349,617],{"stretchy":616},[602,1351,620],{},[608,1353,623],{"stretchy":616},[608,1355,984],{},[628,1357,1358],{},"6",[602,1360,620],{},[632,1362,1363],{"encoding":634},"f''(x) = 6x",[580,1365,1367,1423],{"className":1366,"ariaHidden":640},[639],[580,1368,1370,1373,1405,1408,1411,1414,1417,1420],{"className":1369},[644],[580,1371],{"className":1372,"style":649},[648],[580,1374,1376,1379],{"className":1375},[653],[580,1377,604],{"className":1378,"style":658},[653,657],[580,1380,1382],{"className":1381},[662],[580,1383,1385],{"className":1384},[666],[580,1386,1388],{"className":1387},[670],[580,1389,1391],{"className":1390,"style":675},[674],[580,1392,1393,1396],{"style":678},[580,1394],{"className":1395,"style":683},[682],[580,1397,1399],{"className":1398},[687,688,689,690],[580,1400,1402],{"className":1401},[653,690],[580,1403,697],{"className":1404},[653,690],[580,1406,617],{"className":1407},[701],[580,1409,620],{"className":1410},[653,657],[580,1412,623],{"className":1413},[708],[580,1415],{"className":1416,"style":713},[712],[580,1418,984],{"className":1419},[717],[580,1421],{"className":1422,"style":713},[712],[580,1424,1426,1429,1432],{"className":1425},[644],[580,1427],{"className":1428,"style":727},[648],[580,1430,1358],{"className":1431},[653],[580,1433,620],{"className":1434},[653,657],". Für ",[580,1437,1439,1457],{"className":1438},[583],[580,1440,1442],{"className":1441},[587],[589,1443,1444],{"xmlns":591},[593,1445,1446,1454],{},[596,1447,1448,1450,1452],{},[602,1449,620],{},[608,1451,804],{},[628,1453,630],{},[632,1455,1456],{"encoding":634},"x \u003C 0",[580,1458,1460,1479],{"className":1459,"ariaHidden":640},[639],[580,1461,1463,1467,1470,1473,1476],{"className":1462},[644],[580,1464],{"className":1465,"style":1466},[648],"height:0.5782em;vertical-align:-0.0391em;",[580,1468,620],{"className":1469},[653,657],[580,1471],{"className":1472,"style":713},[712],[580,1474,804],{"className":1475},[717],[580,1477],{"className":1478,"style":713},[712],[580,1480,1482,1485],{"className":1481},[644],[580,1483],{"className":1484,"style":727},[648],[580,1486,630],{"className":1487},[653]," ist ",[580,1490,1492,1523],{"className":1491},[583],[580,1493,1495],{"className":1494},[587],[589,1496,1497],{"xmlns":591},[593,1498,1499,1521],{},[596,1500,1501,1511,1513,1515,1517,1519],{},[599,1502,1503,1505],{},[602,1504,604],{},[596,1506,1507,1509],{},[608,1508,611],{"mathvariant":610},[608,1510,611],{"mathvariant":610},[608,1512,617],{"stretchy":616},[602,1514,620],{},[608,1516,623],{"stretchy":616},[608,1518,804],{},[628,1520,630],{},[632,1522,809],{"encoding":634},[580,1524,1526,1582],{"className":1525,"ariaHidden":640},[639],[580,1527,1529,1532,1564,1567,1570,1573,1576,1579],{"className":1528},[644],[580,1530],{"className":1531,"style":649},[648],[580,1533,1535,1538],{"className":1534},[653],[580,1536,604],{"className":1537,"style":658},[653,657],[580,1539,1541],{"className":1540},[662],[580,1542,1544],{"className":1543},[666],[580,1545,1547],{"className":1546},[670],[580,1548,1550],{"className":1549,"style":675},[674],[580,1551,1552,1555],{"style":678},[580,1553],{"className":1554,"style":683},[682],[580,1556,1558],{"className":1557},[687,688,689,690],[580,1559,1561],{"className":1560},[653,690],[580,1562,697],{"className":1563},[653,690],[580,1565,617],{"className":1566},[701],[580,1568,620],{"className":1569},[653,657],[580,1571,623],{"className":1572},[708],[580,1574],{"className":1575,"style":713},[712],[580,1577,804],{"className":1578},[717],[580,1580],{"className":1581,"style":713},[712],[580,1583,1585,1588],{"className":1584},[644],[580,1586],{"className":1587,"style":727},[648],[580,1589,630],{"className":1590},[653],", z.B. ",[580,1593,1595,1632],{"className":1594},[583],[580,1596,1598],{"className":1597},[587],[589,1599,1600],{"xmlns":591},[593,1601,1602,1629],{},[596,1603,1604,1614,1616,1618,1621,1623,1625,1627],{},[599,1605,1606,1608],{},[602,1607,604],{},[596,1609,1610,1612],{},[608,1611,611],{"mathvariant":610},[608,1613,611],{"mathvariant":610},[608,1615,617],{"stretchy":616},[608,1617,1228],{},[628,1619,1620],{},"1",[608,1622,623],{"stretchy":616},[608,1624,984],{},[608,1626,1228],{},[628,1628,1358],{},[632,1630,1631],{"encoding":634},"f''(-1) = -6",[580,1633,1635,1694],{"className":1634,"ariaHidden":640},[639],[580,1636,1638,1641,1673,1676,1679,1682,1685,1688,1691],{"className":1637},[644],[580,1639],{"className":1640,"style":649},[648],[580,1642,1644,1647],{"className":1643},[653],[580,1645,604],{"className":1646,"style":658},[653,657],[580,1648,1650],{"className":1649},[662],[580,1651,1653],{"className":1652},[666],[580,1654,1656],{"className":1655},[670],[580,1657,1659],{"className":1658,"style":675},[674],[580,1660,1661,1664],{"style":678},[580,1662],{"className":1663,"style":683},[682],[580,1665,1667],{"className":1666},[687,688,689,690],[580,1668,1670],{"className":1669},[653,690],[580,1671,697],{"className":1672},[653,690],[580,1674,617],{"className":1675},[701],[580,1677,1228],{"className":1678},[653],[580,1680,1620],{"className":1681},[653],[580,1683,623],{"className":1684},[708],[580,1686],{"className":1687,"style":713},[712],[580,1689,984],{"className":1690},[717],[580,1692],{"className":1693,"style":713},[712],[580,1695,1697,1701,1704],{"className":1696},[644],[580,1698],{"className":1699,"style":1700},[648],"height:0.7278em;vertical-align:-0.0833em;",[580,1702,1228],{"className":1703},[653],[580,1705,1358],{"className":1706},[653],": Rechtskurve. Für ",[580,1709,1711,1729],{"className":1710},[583],[580,1712,1714],{"className":1713},[587],[589,1715,1716],{"xmlns":591},[593,1717,1718,1726],{},[596,1719,1720,1722,1724],{},[602,1721,620],{},[608,1723,626],{},[628,1725,630],{},[632,1727,1728],{"encoding":634},"x > 0",[580,1730,1732,1750],{"className":1731,"ariaHidden":640},[639],[580,1733,1735,1738,1741,1744,1747],{"className":1734},[644],[580,1736],{"className":1737,"style":1466},[648],[580,1739,620],{"className":1740},[653,657],[580,1742],{"className":1743,"style":713},[712],[580,1745,626],{"className":1746},[717],[580,1748],{"className":1749,"style":713},[712],[580,1751,1753,1756],{"className":1752},[644],[580,1754],{"className":1755,"style":727},[648],[580,1757,630],{"className":1758},[653],[580,1760,1762,1793],{"className":1761},[583],[580,1763,1765],{"className":1764},[587],[589,1766,1767],{"xmlns":591},[593,1768,1769,1791],{},[596,1770,1771,1781,1783,1785,1787,1789],{},[599,1772,1773,1775],{},[602,1774,604],{},[596,1776,1777,1779],{},[608,1778,611],{"mathvariant":610},[608,1780,611],{"mathvariant":610},[608,1782,617],{"stretchy":616},[602,1784,620],{},[608,1786,623],{"stretchy":616},[608,1788,626],{},[628,1790,630],{},[632,1792,635],{"encoding":634},[580,1794,1796,1852],{"className":1795,"ariaHidden":640},[639],[580,1797,1799,1802,1834,1837,1840,1843,1846,1849],{"className":1798},[644],[580,1800],{"className":1801,"style":649},[648],[580,1803,1805,1808],{"className":1804},[653],[580,1806,604],{"className":1807,"style":658},[653,657],[580,1809,1811],{"className":1810},[662],[580,1812,1814],{"className":1813},[666],[580,1815,1817],{"className":1816},[670],[580,1818,1820],{"className":1819,"style":675},[674],[580,1821,1822,1825],{"style":678},[580,1823],{"className":1824,"style":683},[682],[580,1826,1828],{"className":1827},[687,688,689,690],[580,1829,1831],{"className":1830},[653,690],[580,1832,697],{"className":1833},[653,690],[580,1835,617],{"className":1836},[701],[580,1838,620],{"className":1839},[653,657],[580,1841,623],{"className":1842},[708],[580,1844],{"className":1845,"style":713},[712],[580,1847,626],{"className":1848},[717],[580,1850],{"className":1851,"style":713},[712],[580,1853,1855,1858],{"className":1854},[644],[580,1856],{"className":1857,"style":727},[648],[580,1859,630],{"className":1860},[653],[580,1862,1864,1898],{"className":1863},[583],[580,1865,1867],{"className":1866},[587],[589,1868,1869],{"xmlns":591},[593,1870,1871,1895],{},[596,1872,1873,1883,1885,1888,1890,1892],{},[599,1874,1875,1877],{},[602,1876,604],{},[596,1878,1879,1881],{},[608,1880,611],{"mathvariant":610},[608,1882,611],{"mathvariant":610},[608,1884,617],{"stretchy":616},[628,1886,1887],{},"2",[608,1889,623],{"stretchy":616},[608,1891,984],{},[628,1893,1894],{},"12",[632,1896,1897],{"encoding":634},"f''(2) = 12",[580,1899,1901,1957],{"className":1900,"ariaHidden":640},[639],[580,1902,1904,1907,1939,1942,1945,1948,1951,1954],{"className":1903},[644],[580,1905],{"className":1906,"style":649},[648],[580,1908,1910,1913],{"className":1909},[653],[580,1911,604],{"className":1912,"style":658},[653,657],[580,1914,1916],{"className":1915},[662],[580,1917,1919],{"className":1918},[666],[580,1920,1922],{"className":1921},[670],[580,1923,1925],{"className":1924,"style":675},[674],[580,1926,1927,1930],{"style":678},[580,1928],{"className":1929,"style":683},[682],[580,1931,1933],{"className":1932},[687,688,689,690],[580,1934,1936],{"className":1935},[653,690],[580,1937,697],{"className":1938},[653,690],[580,1940,617],{"className":1941},[701],[580,1943,1887],{"className":1944},[653],[580,1946,623],{"className":1947},[708],[580,1949],{"className":1950,"style":713},[712],[580,1952,984],{"className":1953},[717],[580,1955],{"className":1956,"style":713},[712],[580,1958,1960,1963],{"className":1959},[644],[580,1961],{"className":1962,"style":727},[648],[580,1964,1894],{"className":1965},[653],": Linkskurve. Bei ",[580,1968,1970,1988],{"className":1969},[583],[580,1971,1973],{"className":1972},[587],[589,1974,1975],{"xmlns":591},[593,1976,1977,1985],{},[596,1978,1979,1981,1983],{},[602,1980,620],{},[608,1982,984],{},[628,1984,630],{},[632,1986,1987],{"encoding":634},"x = 0",[580,1989,1991,2009],{"className":1990,"ariaHidden":640},[639],[580,1992,1994,1997,2000,2003,2006],{"className":1993},[644],[580,1995],{"className":1996,"style":942},[648],[580,1998,620],{"className":1999},[653,657],[580,2001],{"className":2002,"style":713},[712],[580,2004,984],{"className":2005},[717],[580,2007],{"className":2008,"style":713},[712],[580,2010,2012,2015],{"className":2011},[644],[580,2013],{"className":2014,"style":727},[648],[580,2016,630],{"className":2017},[653]," wechselt die Krümmung, dort liegt die Wendestelle.",[565,2020,2021,1325,2150,2280,2281,2343,2344,2405,2406,2434,2435,1187],{},[580,2022,2024,2061],{"className":2023},[583],[580,2025,2027],{"className":2026},[587],[589,2028,2029],{"xmlns":591},[593,2030,2031,2058],{},[596,2032,2033,2035,2037,2039,2041,2043,2045,2051,2054,2056],{},[602,2034,604],{},[608,2036,617],{"stretchy":616},[602,2038,620],{},[608,2040,623],{"stretchy":616},[608,2042,984],{},[608,2044,1228],{},[599,2046,2047,2049],{},[602,2048,620],{},[628,2050,1887],{},[608,2052,2053],{},"+",[628,2055,1087],{},[602,2057,620],{},[632,2059,2060],{"encoding":634},"f(x) = -x^2 + 4x",[580,2062,2064,2091,2138],{"className":2063,"ariaHidden":640},[639],[580,2065,2067,2070,2073,2076,2079,2082,2085,2088],{"className":2066},[644],[580,2068],{"className":2069,"style":1100},[648],[580,2071,604],{"className":2072,"style":658},[653,657],[580,2074,617],{"className":2075},[701],[580,2077,620],{"className":2078},[653,657],[580,2080,623],{"className":2081},[708],[580,2083],{"className":2084,"style":713},[712],[580,2086,984],{"className":2087},[717],[580,2089],{"className":2090,"style":713},[712],[580,2092,2094,2097,2100,2129,2132,2135],{"className":2093},[644],[580,2095],{"className":2096,"style":1272},[648],[580,2098,1228],{"className":2099},[653],[580,2101,2103,2106],{"className":2102},[653],[580,2104,620],{"className":2105},[653,657],[580,2107,2109],{"className":2108},[662],[580,2110,2112],{"className":2111},[666],[580,2113,2115],{"className":2114},[670],[580,2116,2118],{"className":2117,"style":1128},[674],[580,2119,2120,2123],{"style":678},[580,2121],{"className":2122,"style":683},[682],[580,2124,2126],{"className":2125},[687,688,689,690],[580,2127,1887],{"className":2128},[653,690],[580,2130],{"className":2131,"style":1305},[712],[580,2133,2053],{"className":2134},[1309],[580,2136],{"className":2137,"style":1305},[712],[580,2139,2141,2144,2147],{"className":2140},[644],[580,2142],{"className":2143,"style":727},[648],[580,2145,1087],{"className":2146},[653],[580,2148,620],{"className":2149},[653,657],[580,2151,2153,2191],{"className":2152},[583],[580,2154,2156],{"className":2155},[587],[589,2157,2158],{"xmlns":591},[593,2159,2160,2188],{},[596,2161,2162,2172,2174,2176,2178,2180,2182,2184,2186],{},[599,2163,2164,2166],{},[602,2165,604],{},[596,2167,2168,2170],{},[608,2169,611],{"mathvariant":610},[608,2171,611],{"mathvariant":610},[608,2173,617],{"stretchy":616},[602,2175,620],{},[608,2177,623],{"stretchy":616},[608,2179,984],{},[608,2181,1228],{},[628,2183,1887],{},[608,2185,804],{},[628,2187,630],{},[632,2189,2190],{"encoding":634},"f''(x) = -2 \u003C 0",[580,2192,2194,2250,2271],{"className":2193,"ariaHidden":640},[639],[580,2195,2197,2200,2232,2235,2238,2241,2244,2247],{"className":2196},[644],[580,2198],{"className":2199,"style":649},[648],[580,2201,2203,2206],{"className":2202},[653],[580,2204,604],{"className":2205,"style":658},[653,657],[580,2207,2209],{"className":2208},[662],[580,2210,2212],{"className":2211},[666],[580,2213,2215],{"className":2214},[670],[580,2216,2218],{"className":2217,"style":675},[674],[580,2219,2220,2223],{"style":678},[580,2221],{"className":2222,"style":683},[682],[580,2224,2226],{"className":2225},[687,688,689,690],[580,2227,2229],{"className":2228},[653,690],[580,2230,697],{"className":2231},[653,690],[580,2233,617],{"className":2234},[701],[580,2236,620],{"className":2237},[653,657],[580,2239,623],{"className":2240},[708],[580,2242],{"className":2243,"style":713},[712],[580,2245,984],{"className":2246},[717],[580,2248],{"className":2249,"style":713},[712],[580,2251,2253,2256,2259,2262,2265,2268],{"className":2252},[644],[580,2254],{"className":2255,"style":1700},[648],[580,2257,1228],{"className":2258},[653],[580,2260,1887],{"className":2261},[653],[580,2263],{"className":2264,"style":713},[712],[580,2266,804],{"className":2267},[717],[580,2269],{"className":2270,"style":713},[712],[580,2272,2274,2277],{"className":2273},[644],[580,2275],{"className":2276,"style":727},[648],[580,2278,630],{"className":2279},[653]," ist überall eine Rechtskurve. Die Sekante durch ",[580,2282,2284,2307],{"className":2283},[583],[580,2285,2287],{"className":2286},[587],[589,2288,2289],{"xmlns":591},[593,2290,2291,2304],{},[596,2292,2293,2295,2297,2300,2302],{},[608,2294,617],{"stretchy":616},[628,2296,1620],{},[608,2298,2299],{},"∣",[628,2301,1225],{},[608,2303,623],{"stretchy":616},[632,2305,2306],{"encoding":634},"(1 \\mid 3)",[580,2308,2310,2331],{"className":2309,"ariaHidden":640},[639],[580,2311,2313,2316,2319,2322,2325,2328],{"className":2312},[644],[580,2314],{"className":2315,"style":1100},[648],[580,2317,617],{"className":2318},[701],[580,2320,1620],{"className":2321},[653],[580,2323],{"className":2324,"style":713},[712],[580,2326,2299],{"className":2327},[717],[580,2329],{"className":2330,"style":713},[712],[580,2332,2334,2337,2340],{"className":2333},[644],[580,2335],{"className":2336,"style":1100},[648],[580,2338,1225],{"className":2339},[653],[580,2341,623],{"className":2342},[708]," und ",[580,2345,2347,2369],{"className":2346},[583],[580,2348,2350],{"className":2349},[587],[589,2351,2352],{"xmlns":591},[593,2353,2354,2366],{},[596,2355,2356,2358,2360,2362,2364],{},[608,2357,617],{"stretchy":616},[628,2359,1225],{},[608,2361,2299],{},[628,2363,1225],{},[608,2365,623],{"stretchy":616},[632,2367,2368],{"encoding":634},"(3 \\mid 3)",[580,2370,2372,2393],{"className":2371,"ariaHidden":640},[639],[580,2373,2375,2378,2381,2384,2387,2390],{"className":2374},[644],[580,2376],{"className":2377,"style":1100},[648],[580,2379,617],{"className":2380},[701],[580,2382,1225],{"className":2383},[653],[580,2385],{"className":2386,"style":713},[712],[580,2388,2299],{"className":2389},[717],[580,2391],{"className":2392,"style":713},[712],[580,2394,2396,2399,2402],{"className":2395},[644],[580,2397],{"className":2398,"style":1100},[648],[580,2400,1225],{"className":2401},[653],[580,2403,623],{"className":2404},[708]," verläuft auf der Höhe ",[580,2407,2409,2422],{"className":2408},[583],[580,2410,2412],{"className":2411},[587],[589,2413,2414],{"xmlns":591},[593,2415,2416,2420],{},[596,2417,2418],{},[628,2419,1225],{},[632,2421,1225],{"encoding":634},[580,2423,2425],{"className":2424,"ariaHidden":640},[639],[580,2426,2428,2431],{"className":2427},[644],[580,2429],{"className":2430,"style":727},[648],[580,2432,1225],{"className":2433},[653],", der Graph liegt dazwischen darüber, z.B. ",[580,2436,2438,2462],{"className":2437},[583],[580,2439,2441],{"className":2440},[587],[589,2442,2443],{"xmlns":591},[593,2444,2445,2459],{},[596,2446,2447,2449,2451,2453,2455,2457],{},[602,2448,604],{},[608,2450,617],{"stretchy":616},[628,2452,1887],{},[608,2454,623],{"stretchy":616},[608,2456,984],{},[628,2458,1087],{},[632,2460,2461],{"encoding":634},"f(2) = 4",[580,2463,2465,2492],{"className":2464,"ariaHidden":640},[639],[580,2466,2468,2471,2474,2477,2480,2483,2486,2489],{"className":2467},[644],[580,2469],{"className":2470,"style":1100},[648],[580,2472,604],{"className":2473,"style":658},[653,657],[580,2475,617],{"className":2476},[701],[580,2478,1887],{"className":2479},[653],[580,2481,623],{"className":2482},[708],[580,2484],{"className":2485,"style":713},[712],[580,2487,984],{"className":2488},[717],[580,2490],{"className":2491,"style":713},[712],[580,2493,2495,2498],{"className":2494},[644],[580,2496],{"className":2497,"style":727},[648],[580,2499,1087],{"className":2500},[653],[565,2502,2503,1325,2598,2736,2737,2840,2841,2943,2944,3037,3038,3066],{},[580,2504,2506,2533],{"className":2505},[583],[580,2507,2509],{"className":2508},[587],[589,2510,2511],{"xmlns":591},[593,2512,2513,2531],{},[596,2514,2515,2517,2519,2521,2523,2525],{},[602,2516,604],{},[608,2518,617],{"stretchy":616},[602,2520,620],{},[608,2522,623],{"stretchy":616},[608,2524,984],{},[599,2526,2527,2529],{},[602,2528,620],{},[628,2530,1087],{},[632,2532,1090],{"encoding":634},[580,2534,2536,2563],{"className":2535,"ariaHidden":640},[639],[580,2537,2539,2542,2545,2548,2551,2554,2557,2560],{"className":2538},[644],[580,2540],{"className":2541,"style":1100},[648],[580,2543,604],{"className":2544,"style":658},[653,657],[580,2546,617],{"className":2547},[701],[580,2549,620],{"className":2550},[653,657],[580,2552,623],{"className":2553},[708],[580,2555],{"className":2556,"style":713},[712],[580,2558,984],{"className":2559},[717],[580,2561],{"className":2562,"style":713},[712],[580,2564,2566,2569],{"className":2565},[644],[580,2567],{"className":2568,"style":1128},[648],[580,2570,2572,2575],{"className":2571},[653],[580,2573,620],{"className":2574},[653,657],[580,2576,2578],{"className":2577},[662],[580,2579,2581],{"className":2580},[666],[580,2582,2584],{"className":2583},[670],[580,2585,2587],{"className":2586,"style":1128},[674],[580,2588,2589,2592],{"style":678},[580,2590],{"className":2591,"style":683},[682],[580,2593,2595],{"className":2594},[687,688,689,690],[580,2596,1087],{"className":2597},[653,690],[580,2599,2601,2639],{"className":2600},[583],[580,2602,2604],{"className":2603},[587],[589,2605,2606],{"xmlns":591},[593,2607,2608,2636],{},[596,2609,2610,2620,2622,2624,2626,2628,2630],{},[599,2611,2612,2614],{},[602,2613,604],{},[596,2615,2616,2618],{},[608,2617,611],{"mathvariant":610},[608,2619,611],{"mathvariant":610},[608,2621,617],{"stretchy":616},[602,2623,620],{},[608,2625,623],{"stretchy":616},[608,2627,984],{},[628,2629,1894],{},[599,2631,2632,2634],{},[602,2633,620],{},[628,2635,1887],{},[632,2637,2638],{"encoding":634},"f''(x) = 12x^2",[580,2640,2642,2698],{"className":2641,"ariaHidden":640},[639],[580,2643,2645,2648,2680,2683,2686,2689,2692,2695],{"className":2644},[644],[580,2646],{"className":2647,"style":649},[648],[580,2649,2651,2654],{"className":2650},[653],[580,2652,604],{"className":2653,"style":658},[653,657],[580,2655,2657],{"className":2656},[662],[580,2658,2660],{"className":2659},[666],[580,2661,2663],{"className":2662},[670],[580,2664,2666],{"className":2665,"style":675},[674],[580,2667,2668,2671],{"style":678},[580,2669],{"className":2670,"style":683},[682],[580,2672,2674],{"className":2673},[687,688,689,690],[580,2675,2677],{"className":2676},[653,690],[580,2678,697],{"className":2679},[653,690],[580,2681,617],{"className":2682},[701],[580,2684,620],{"className":2685},[653,657],[580,2687,623],{"className":2688},[708],[580,2690],{"className":2691,"style":713},[712],[580,2693,984],{"className":2694},[717],[580,2696],{"className":2697,"style":713},[712],[580,2699,2701,2704,2707],{"className":2700},[644],[580,2702],{"className":2703,"style":1128},[648],[580,2705,1894],{"className":2706},[653],[580,2708,2710,2713],{"className":2709},[653],[580,2711,620],{"className":2712},[653,657],[580,2714,2716],{"className":2715},[662],[580,2717,2719],{"className":2718},[666],[580,2720,2722],{"className":2721},[670],[580,2723,2725],{"className":2724,"style":1128},[674],[580,2726,2727,2730],{"style":678},[580,2728],{"className":2729,"style":683},[682],[580,2731,2733],{"className":2732},[687,688,689,690],[580,2734,1887],{"className":2735},[653,690],". Es gilt ",[580,2738,2740,2772],{"className":2739},[583],[580,2741,2743],{"className":2742},[587],[589,2744,2745],{"xmlns":591},[593,2746,2747,2769],{},[596,2748,2749,2759,2761,2763,2765,2767],{},[599,2750,2751,2753],{},[602,2752,604],{},[596,2754,2755,2757],{},[608,2756,611],{"mathvariant":610},[608,2758,611],{"mathvariant":610},[608,2760,617],{"stretchy":616},[628,2762,630],{},[608,2764,623],{"stretchy":616},[608,2766,984],{},[628,2768,630],{},[632,2770,2771],{"encoding":634},"f''(0) = 0",[580,2773,2775,2831],{"className":2774,"ariaHidden":640},[639],[580,2776,2778,2781,2813,2816,2819,2822,2825,2828],{"className":2777},[644],[580,2779],{"className":2780,"style":649},[648],[580,2782,2784,2787],{"className":2783},[653],[580,2785,604],{"className":2786,"style":658},[653,657],[580,2788,2790],{"className":2789},[662],[580,2791,2793],{"className":2792},[666],[580,2794,2796],{"className":2795},[670],[580,2797,2799],{"className":2798,"style":675},[674],[580,2800,2801,2804],{"style":678},[580,2802],{"className":2803,"style":683},[682],[580,2805,2807],{"className":2806},[687,688,689,690],[580,2808,2810],{"className":2809},[653,690],[580,2811,697],{"className":2812},[653,690],[580,2814,617],{"className":2815},[701],[580,2817,630],{"className":2818},[653],[580,2820,623],{"className":2821},[708],[580,2823],{"className":2824,"style":713},[712],[580,2826,984],{"className":2827},[717],[580,2829],{"className":2830,"style":713},[712],[580,2832,2834,2837],{"className":2833},[644],[580,2835],{"className":2836,"style":727},[648],[580,2838,630],{"className":2839},[653],", aber ",[580,2842,2844,2875],{"className":2843},[583],[580,2845,2847],{"className":2846},[587],[589,2848,2849],{"xmlns":591},[593,2850,2851,2873],{},[596,2852,2853,2863,2865,2867,2869,2871],{},[599,2854,2855,2857],{},[602,2856,604],{},[596,2858,2859,2861],{},[608,2860,611],{"mathvariant":610},[608,2862,611],{"mathvariant":610},[608,2864,617],{"stretchy":616},[602,2866,620],{},[608,2868,623],{"stretchy":616},[608,2870,626],{},[628,2872,630],{},[632,2874,635],{"encoding":634},[580,2876,2878,2934],{"className":2877,"ariaHidden":640},[639],[580,2879,2881,2884,2916,2919,2922,2925,2928,2931],{"className":2880},[644],[580,2882],{"className":2883,"style":649},[648],[580,2885,2887,2890],{"className":2886},[653],[580,2888,604],{"className":2889,"style":658},[653,657],[580,2891,2893],{"className":2892},[662],[580,2894,2896],{"className":2895},[666],[580,2897,2899],{"className":2898},[670],[580,2900,2902],{"className":2901,"style":675},[674],[580,2903,2904,2907],{"style":678},[580,2905],{"className":2906,"style":683},[682],[580,2908,2910],{"className":2909},[687,688,689,690],[580,2911,2913],{"className":2912},[653,690],[580,2914,697],{"className":2915},[653,690],[580,2917,617],{"className":2918},[701],[580,2920,620],{"className":2921},[653,657],[580,2923,623],{"className":2924},[708],[580,2926],{"className":2927,"style":713},[712],[580,2929,626],{"className":2930},[717],[580,2932],{"className":2933,"style":713},[712],[580,2935,2937,2940],{"className":2936},[644],[580,2938],{"className":2939,"style":727},[648],[580,2941,630],{"className":2942},[653]," für alle ",[580,2945,2947,2966],{"className":2946},[583],[580,2948,2950],{"className":2949},[587],[589,2951,2952],{"xmlns":591},[593,2953,2954,2963],{},[596,2955,2956,2958,2961],{},[602,2957,620],{},[608,2959,2960],{"mathvariant":610},"≠",[628,2962,630],{},[632,2964,2965],{"encoding":634},"x \\neq 0",[580,2967,2969,3028],{"className":2968,"ariaHidden":640},[639],[580,2970,2972,2976,2979,2982,3025],{"className":2971},[644],[580,2973],{"className":2974,"style":2975},[648],"height:0.8889em;vertical-align:-0.1944em;",[580,2977,620],{"className":2978},[653,657],[580,2980],{"className":2981,"style":713},[712],[580,2983,2985,3018,3022],{"className":2984},[717],[580,2986,2988],{"className":2987},[717],[580,2989,2992],{"className":2990},[653,2991],"vbox",[580,2993,2996],{"className":2994},[2995],"thinbox",[580,2997,3000,3003,3014],{"className":2998},[2999],"rlap",[580,3001],{"className":3002,"style":2975},[648],[580,3004,3007],{"className":3005},[3006],"inner",[580,3008,3010],{"className":3009},[653],[580,3011,3013],{"className":3012},[717],"",[580,3015],{"className":3016},[3017],"fix",[580,3019],{"className":3020},[712,3021],"nobreak",[580,3023,984],{"className":3024},[717],[580,3026],{"className":3027,"style":713},[712],[580,3029,3031,3034],{"className":3030},[644],[580,3032],{"className":3033,"style":727},[648],[580,3035,630],{"className":3036},[653],". Der Graph ist überall eine Linkskurve, ",[580,3039,3041,3054],{"className":3040},[583],[580,3042,3044],{"className":3043},[587],[589,3045,3046],{"xmlns":591},[593,3047,3048,3052],{},[596,3049,3050],{},[628,3051,630],{},[632,3053,630],{"encoding":634},[580,3055,3057],{"className":3056,"ariaHidden":640},[639],[580,3058,3060,3063],{"className":3059},[644],[580,3061],{"className":3062,"style":727},[648],[580,3064,630],{"className":3065},[653]," ist keine Wendestelle.",[560,3068,3070],{"id":3069},"typische-fehler","Typische Fehler",[574,3072,3073,3318,3424],{},[577,3074,3075,3076,3135,3136,3186,3187,3251,3252,1187],{},"Krümmung und Monotonie verwechseln. Eine Linkskurve kann fallen, z.B. ",[580,3077,3079,3097],{"className":3078},[583],[580,3080,3082],{"className":3081},[587],[589,3083,3084],{"xmlns":591},[593,3085,3086,3094],{},[596,3087,3088],{},[599,3089,3090,3092],{},[602,3091,620],{},[628,3093,1887],{},[632,3095,3096],{"encoding":634},"x^2",[580,3098,3100],{"className":3099,"ariaHidden":640},[639],[580,3101,3103,3106],{"className":3102},[644],[580,3104],{"className":3105,"style":1128},[648],[580,3107,3109,3112],{"className":3108},[653],[580,3110,620],{"className":3111},[653,657],[580,3113,3115],{"className":3114},[662],[580,3116,3118],{"className":3117},[666],[580,3119,3121],{"className":3120},[670],[580,3122,3124],{"className":3123,"style":1128},[674],[580,3125,3126,3129],{"style":678},[580,3127],{"className":3128,"style":683},[682],[580,3130,3132],{"className":3131},[687,688,689,690],[580,3133,1887],{"className":3134},[653,690]," für ",[580,3137,3139,3156],{"className":3138},[583],[580,3140,3142],{"className":3141},[587],[589,3143,3144],{"xmlns":591},[593,3145,3146,3154],{},[596,3147,3148,3150,3152],{},[602,3149,620],{},[608,3151,804],{},[628,3153,630],{},[632,3155,1456],{"encoding":634},[580,3157,3159,3177],{"className":3158,"ariaHidden":640},[639],[580,3160,3162,3165,3168,3171,3174],{"className":3161},[644],[580,3163],{"className":3164,"style":1466},[648],[580,3166,620],{"className":3167},[653,657],[580,3169],{"className":3170,"style":713},[712],[580,3172,804],{"className":3173},[717],[580,3175],{"className":3176,"style":713},[712],[580,3178,3180,3183],{"className":3179},[644],[580,3181],{"className":3182,"style":727},[648],[580,3184,630],{"className":3185},[653],". Für die Monotonie ist ",[580,3188,3190,3209],{"className":3189},[583],[580,3191,3193],{"className":3192},[587],[589,3194,3195],{"xmlns":591},[593,3196,3197,3206],{},[596,3198,3199],{},[599,3200,3201,3203],{},[602,3202,604],{},[608,3204,611],{"mathvariant":610,"lspace":3205,"rspace":3205},"0em",[632,3207,3208],{"encoding":634},"f'",[580,3210,3212],{"className":3211,"ariaHidden":640},[639],[580,3213,3215,3219],{"className":3214},[644],[580,3216],{"className":3217,"style":3218},[648],"height:0.9463em;vertical-align:-0.1944em;",[580,3220,3222,3225],{"className":3221},[653],[580,3223,604],{"className":3224,"style":658},[653,657],[580,3226,3228],{"className":3227},[662],[580,3229,3231],{"className":3230},[666],[580,3232,3234],{"className":3233},[670],[580,3235,3237],{"className":3236,"style":675},[674],[580,3238,3239,3242],{"style":678},[580,3240],{"className":3241,"style":683},[682],[580,3243,3245],{"className":3244},[687,688,689,690],[580,3246,3248],{"className":3247},[653,690],[580,3249,611],{"className":3250},[653,690]," zuständig, für die Krümmung ",[580,3253,3255,3277],{"className":3254},[583],[580,3256,3258],{"className":3257},[587],[589,3259,3260],{"xmlns":591},[593,3261,3262,3274],{},[596,3263,3264],{},[599,3265,3266,3268],{},[602,3267,604],{},[596,3269,3270,3272],{},[608,3271,611],{"mathvariant":610},[608,3273,611],{"mathvariant":610},[632,3275,3276],{"encoding":634},"f''",[580,3278,3280],{"className":3279,"ariaHidden":640},[639],[580,3281,3283,3286],{"className":3282},[644],[580,3284],{"className":3285,"style":3218},[648],[580,3287,3289,3292],{"className":3288},[653],[580,3290,604],{"className":3291,"style":658},[653,657],[580,3293,3295],{"className":3294},[662],[580,3296,3298],{"className":3297},[666],[580,3299,3301],{"className":3300},[670],[580,3302,3304],{"className":3303,"style":675},[674],[580,3305,3306,3309],{"style":678},[580,3307],{"className":3308,"style":683},[682],[580,3310,3312],{"className":3311},[687,688,689,690],[580,3313,3315],{"className":3314},[653,690],[580,3316,697],{"className":3317},[653,690],[577,3319,3320,3321,3423],{},"Links- und Rechtskurve vertauschen. Merkhilfe: ",[580,3322,3324,3355],{"className":3323},[583],[580,3325,3327],{"className":3326},[587],[589,3328,3329],{"xmlns":591},[593,3330,3331,3353],{},[596,3332,3333,3343,3345,3347,3349,3351],{},[599,3334,3335,3337],{},[602,3336,604],{},[596,3338,3339,3341],{},[608,3340,611],{"mathvariant":610},[608,3342,611],{"mathvariant":610},[608,3344,617],{"stretchy":616},[602,3346,620],{},[608,3348,623],{"stretchy":616},[608,3350,626],{},[628,3352,630],{},[632,3354,635],{"encoding":634},[580,3356,3358,3414],{"className":3357,"ariaHidden":640},[639],[580,3359,3361,3364,3396,3399,3402,3405,3408,3411],{"className":3360},[644],[580,3362],{"className":3363,"style":649},[648],[580,3365,3367,3370],{"className":3366},[653],[580,3368,604],{"className":3369,"style":658},[653,657],[580,3371,3373],{"className":3372},[662],[580,3374,3376],{"className":3375},[666],[580,3377,3379],{"className":3378},[670],[580,3380,3382],{"className":3381,"style":675},[674],[580,3383,3384,3387],{"style":678},[580,3385],{"className":3386,"style":683},[682],[580,3388,3390],{"className":3389},[687,688,689,690],[580,3391,3393],{"className":3392},[653,690],[580,3394,697],{"className":3395},[653,690],[580,3397,617],{"className":3398},[701],[580,3400,620],{"className":3401},[653,657],[580,3403,623],{"className":3404},[708],[580,3406],{"className":3407,"style":713},[712],[580,3409,626],{"className":3410},[717],[580,3412],{"className":3413,"style":713},[712],[580,3415,3417,3420],{"className":3416},[644],[580,3418],{"className":3419,"style":727},[648],[580,3421,630],{"className":3422},[653]," wie bei der nach oben offenen Normalparabel, Linkskurve.",[577,3425,3426,3427,3578,3579,3644],{},"Aus ",[580,3428,3430,3467],{"className":3429},[583],[580,3431,3433],{"className":3432},[587],[589,3434,3435],{"xmlns":591},[593,3436,3437,3464],{},[596,3438,3439,3449,3451,3458,3460,3462],{},[599,3440,3441,3443],{},[602,3442,604],{},[596,3444,3445,3447],{},[608,3446,611],{"mathvariant":610},[608,3448,611],{"mathvariant":610},[608,3450,617],{"stretchy":616},[3452,3453,3454,3456],"msub",{},[602,3455,620],{},[628,3457,630],{},[608,3459,623],{"stretchy":616},[608,3461,984],{},[628,3463,630],{},[632,3465,3466],{"encoding":634},"f''(x_0) = 0",[580,3468,3470,3569],{"className":3469,"ariaHidden":640},[639],[580,3471,3473,3476,3508,3511,3557,3560,3563,3566],{"className":3472},[644],[580,3474],{"className":3475,"style":649},[648],[580,3477,3479,3482],{"className":3478},[653],[580,3480,604],{"className":3481,"style":658},[653,657],[580,3483,3485],{"className":3484},[662],[580,3486,3488],{"className":3487},[666],[580,3489,3491],{"className":3490},[670],[580,3492,3494],{"className":3493,"style":675},[674],[580,3495,3496,3499],{"style":678},[580,3497],{"className":3498,"style":683},[682],[580,3500,3502],{"className":3501},[687,688,689,690],[580,3503,3505],{"className":3504},[653,690],[580,3506,697],{"className":3507},[653,690],[580,3509,617],{"className":3510},[701],[580,3512,3514,3517],{"className":3513},[653],[580,3515,620],{"className":3516},[653,657],[580,3518,3520],{"className":3519},[662],[580,3521,3524,3548],{"className":3522},[666,3523],"vlist-t2",[580,3525,3527,3543],{"className":3526},[670],[580,3528,3531],{"className":3529,"style":3530},[674],"height:0.3011em;",[580,3532,3534,3537],{"style":3533},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[580,3535],{"className":3536,"style":683},[682],[580,3538,3540],{"className":3539},[687,688,689,690],[580,3541,630],{"className":3542},[653,690],[580,3544,3547],{"className":3545},[3546],"vlist-s","​",[580,3549,3551],{"className":3550},[670],[580,3552,3555],{"className":3553,"style":3554},[674],"height:0.15em;",[580,3556],{},[580,3558,623],{"className":3559},[708],[580,3561],{"className":3562,"style":713},[712],[580,3564,984],{"className":3565},[717],[580,3567],{"className":3568,"style":713},[712],[580,3570,3572,3575],{"className":3571},[644],[580,3573],{"className":3574,"style":727},[648],[580,3576,630],{"className":3577},[653]," auf einen Krümmungswechsel schließen, ohne das Vorzeichen von ",[580,3580,3582,3603],{"className":3581},[583],[580,3583,3585],{"className":3584},[587],[589,3586,3587],{"xmlns":591},[593,3588,3589,3601],{},[596,3590,3591],{},[599,3592,3593,3595],{},[602,3594,604],{},[596,3596,3597,3599],{},[608,3598,611],{"mathvariant":610},[608,3600,611],{"mathvariant":610},[632,3602,3276],{"encoding":634},[580,3604,3606],{"className":3605,"ariaHidden":640},[639],[580,3607,3609,3612],{"className":3608},[644],[580,3610],{"className":3611,"style":3218},[648],[580,3613,3615,3618],{"className":3614},[653],[580,3616,604],{"className":3617,"style":658},[653,657],[580,3619,3621],{"className":3620},[662],[580,3622,3624],{"className":3623},[666],[580,3625,3627],{"className":3626},[670],[580,3628,3630],{"className":3629,"style":675},[674],[580,3631,3632,3635],{"style":678},[580,3633],{"className":3634,"style":683},[682],[580,3636,3638],{"className":3637},[687,688,689,690],[580,3639,3641],{"className":3640},[653,690],[580,3642,697],{"className":3643},[653,690]," links und rechts zu prüfen.",{"title":3646,"searchDepth":3647,"depth":3647,"links":3648},"",2,[3649,3650,3651],{"id":562,"depth":3647,"text":563},{"id":1193,"depth":3647,"text":1194},{"id":3069,"depth":3647,"text":3070},[3653,3655,3658,3660,3662,3664,3666,3668],{"lhs":3654,"rhs":630},"round(((x+1e-4)^3 - 3*(x+1e-4) - 2*(x^3 - 3*x) + (x-1e-4)^3 - 3*(x-1e-4))\u002F(1e-8) - 6*x, 4)",{"lhs":3656,"rhs":3657},"6*(-1)","-6",{"lhs":3659,"rhs":1894},"6*2",{"lhs":3661,"rhs":1225},"-(1^2) + 4*1",{"lhs":3663,"rhs":1225},"-(3^2) + 4*3",{"lhs":3665,"rhs":1087},"-(2^2) + 4*2",{"lhs":3667,"rhs":630},"12*0^2",{"lhs":3669,"rhs":630},"round(((x+1e-4)^4 - 2*x^4 + (x-1e-4)^4)\u002F(1e-8) - 12*x^2, 4)","f \\text{ zweimal differenzierbar auf dem Intervall } I","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">zweimal differenzierbar auf dem Intervall\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>I\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">I\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">I\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":3675,"x":3676,"y":3679,"unit":3682,"unitY":3683,"axes":3684,"items":3685},"Graph von f(x) = x³ − 3x, für x \u003C 0 eine Rechtskurve, für x > 0 eine Linkskurve, mit dem Wendepunkt W(0 | 0)",[3677,3678],-2.4,2.4,[3680,3681],-3.4,3.4,50,34,true,[3686,3689,3697,3703],{"type":3687,"expr":3688},"function","x^3 - 3*x",{"type":3690,"at":3691,"label":3693,"offset":3694},"point",[3692,3692],0,"W",[3695,3696],10,14,{"type":3698,"at":3699,"text":3702},"label",[3700,3701],-1.3,3,"f″ \u003C 0",{"type":3698,"at":3704,"text":3707},[3705,3706],1.3,-3,"f″ > 0","\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 240 231.2\" width=\"240\" height=\"231.2\" role=\"img\" aria-label=\"Graph von f(x) = x³ − 3x, für x < 0 eine Rechtskurve, für x > 0 eine Linkskurve, mit dem 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text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">W\u003C\u002Ftext>\u003Ctext x=\"55\" y=\"13.6\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">f″ < 0\u003C\u002Ftext>\u003Ctext x=\"185\" y=\"217.6\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">f″ > 0\u003C\u002Ftext>\u003C\u002Fsvg>","curvature-behavior","\\begin{aligned} f'' > 0 \\text{ auf } I &\\;\\Rightarrow\\; \\text{Linkskurve} \\\\ f'' \u003C 0 \\text{ auf } I &\\;\\Rightarrow\\; \\text{Rechtskurve} \\end{aligned}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmtable rowspacing=\"0.25em\" columnalign=\"right left\" columnspacing=\"0em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"true\">\u003Cmrow>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmo mathvariant=\"normal\">′\u003C\u002Fmo>\u003Cmo mathvariant=\"normal\">′\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmtext> auf \u003C\u002Fmtext>\u003Cmi>I\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"true\">\u003Cmrow>\u003Cmrow>\u003C\u002Fmrow>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>Linkskurve\u003C\u002Fmtext>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"true\">\u003Cmrow>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmo mathvariant=\"normal\">′\u003C\u002Fmo>\u003Cmo mathvariant=\"normal\">′\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo><\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmtext> auf \u003C\u002Fmtext>\u003Cmi>I\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"true\">\u003Cmrow>\u003Cmrow>\u003C\u002Fmrow>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>Rechtskurve\u003C\u002Fmtext>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003Cannotation encoding=\"application\u002Fx-tex\">\\begin{aligned} f'' > 0 \\text{ auf } I &\\;\\Rightarrow\\; \\text{Linkskurve} \\\\ f'' < 0 \\text{ auf } I &\\;\\Rightarrow\\; \\text{Rechtskurve} \\end{aligned}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.7em;vertical-align:-1.1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mtable\">\u003Cspan class=\"col-align-r\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6em;\">\u003Cspan style=\"top:-3.76em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\"> auf \u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">I\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.26em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\"><\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\"> auf \u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">I\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.1em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"col-align-l\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6em;\">\u003Cspan style=\"top:-3.76em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">Linkskurve\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan 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style=\"height:1.1em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},"\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fkruemmungsverhalten",[],[3717,3718,3719,3720],"inflection-condition","extremum-condition","curve-sketching","monotonicity",{"title":548,"description":3646},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fkruemmungsverhalten","Ist die zweite Ableitung auf einem Intervall positiv, ist der Graph dort eine Linkskurve, ist sie negativ, eine Rechtskurve.",[3725,3726,3727,3728,770],"krümmung","linkskurve","rechtskurve","zweite ableitung",[3730,3733],{"symbol":3276,"meaning":3731,"symbolHtml":3732},"zweite Ableitung von f","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":746,"meaning":3734,"symbolHtml":3735},"Intervall, auf dem das Krümmungsverhalten untersucht wird","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>I\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">I\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">I\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","og7ZACHvSb5RmxaqKrv3V-ttzPq2OjvyIBk7C4Jl1iU",[3738,3741,3744,3747],{"stem":3739,"title":3740,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbedingungen-extremstellen","Bedingungen für Extremstellen",{"stem":3742,"title":3743,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbedingungen-wendestellen","Bedingungen für Wendestellen",{"stem":3745,"title":3746,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fkurvendiskussion","Kurvendiskussion",{"stem":3748,"title":3749,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fmonotoniekriterium","Monotoniekriterium",1793824282582]