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0",[573,1138,1140,1233],{"className":1139,"ariaHidden":640},[639],[573,1141,1143,1146,1178,1181,1221,1224,1227,1230],{"className":1142},[644],[573,1144],{"className":1145,"style":839},[648],[573,1147,1149,1152],{"className":1148},[653],[573,1150,602],{"className":1151,"style":658},[653,657],[573,1153,1155],{"className":1154},[662],[573,1156,1158],{"className":1157},[666],[573,1159,1161],{"className":1160},[670],[573,1162,1164],{"className":1163,"style":858},[674],[573,1165,1166,1169],{"style":861},[573,1167],{"className":1168,"style":683},[682],[573,1170,1172],{"className":1171},[687,688,689,690],[573,1173,1175],{"className":1174},[653,690],[573,1176,874],{"className":1177},[653,690],[573,1179,612],{"className":1180},[700],[573,1182,1184,1187],{"className":1183},[653],[573,1185,618],{"className":1186},[653,657],[573,1188,1190],{"className":1189},[662],[573,1191,1193,1213],{"className":1192},[666,713],[573,1194,1196,1210],{"className":1195},[670],[573,1197,1199],{"className":1198,"style":720},[674],[573,1200,1201,1204],{"style":723},[573,1202],{"className":1203,"style":683},[682],[573,1205,1207],{"className":1206},[687,688,689,690],[573,1208,622],{"className":1209},[653,690],[573,1211,737],{"className":1212},[736],[573,1214,1216],{"className":1215},[670],[573,1217,1219],{"className":1218,"style":744},[674],[573,1220],{},[573,1222,625],{"className":1223},[750],[573,1225],{"className":1226,"style":755},[754],[573,1228,628],{"className":1229},[759],[573,1231],{"className":1232,"style":755},[754],[573,1234,1236,1239],{"className":1235},[644],[573,1237],{"className":1238,"style":769},[648],[573,1240,622],{"className":1241},[653],": Keine Aussage. Dann untersucht man den ",[569,1244,1245],{},"Vorzeichenwechsel"," von ",[573,1248,1250,1268],{"className":1249},[580],[573,1251,1253],{"className":1252},[584],[586,1254,1255],{"xmlns":588},[591,1256,1257,1265],{},[594,1258,1259],{},[597,1260,1261,1263],{},[600,1262,602],{},[604,1264,608],{"mathvariant":606,"lspace":607,"rspace":607},[632,1266,1267],{"encoding":634},"f'",[573,1269,1271],{"className":1270,"ariaHidden":640},[639],[573,1272,1274,1278],{"className":1273},[644],[573,1275],{"className":1276,"style":1277},[648],"height:0.9463em;vertical-align:-0.1944em;",[573,1279,1281,1284],{"className":1280},[653],[573,1282,602],{"className":1283,"style":658},[653,657],[573,1285,1287],{"className":1286},[662],[573,1288,1290],{"className":1289},[666],[573,1291,1293],{"className":1292},[670],[573,1294,1296],{"className":1295,"style":858},[674],[573,1297,1298,1301],{"style":861},[573,1299],{"className":1300,"style":683},[682],[573,1302,1304],{"className":1303},[687,688,689,690],[573,1305,1307],{"className":1306},[653,690],[573,1308,608],{"className":1309},[653,690]," an der Stelle ",[573,1312,1314,1332],{"className":1313},[580],[573,1315,1317],{"className":1316},[584],[586,1318,1319],{"xmlns":588},[591,1320,1321,1329],{},[594,1322,1323],{},[614,1324,1325,1327],{},[600,1326,618],{},[620,1328,622],{},[632,1330,1331],{"encoding":634},"x_0",[573,1333,1335],{"className":1334,"ariaHidden":640},[639],[573,1336,1338,1342],{"className":1337},[644],[573,1339],{"className":1340,"style":1341},[648],"height:0.5806em;vertical-align:-0.15em;",[573,1343,1345,1348],{"className":1344},[653],[573,1346,618],{"className":1347},[653,657],[573,1349,1351],{"className":1350},[662],[573,1352,1354,1374],{"className":1353},[666,713],[573,1355,1357,1371],{"className":1356},[670],[573,1358,1360],{"className":1359,"style":720},[674],[573,1361,1362,1365],{"style":723},[573,1363],{"className":1364,"style":683},[682],[573,1366,1368],{"className":1367},[687,688,689,690],[573,1369,622],{"className":1370},[653,690],[573,1372,737],{"className":1373},[736],[573,1375,1377],{"className":1376},[670],[573,1378,1380],{"className":1379,"style":744},[674],[573,1381],{},": von ",[573,1384,1386,1400],{"className":1385},[580],[573,1387,1389],{"className":1388},[584],[586,1390,1391],{"xmlns":588},[591,1392,1393,1398],{},[594,1394,1395],{},[604,1396,1397],{},"+",[632,1399,1397],{"encoding":634},[573,1401,1403],{"className":1402,"ariaHidden":640},[639],[573,1404,1406,1410],{"className":1405},[644],[573,1407],{"className":1408,"style":1409},[648],"height:0.6667em;vertical-align:-0.0833em;",[573,1411,1397],{"className":1412},[653]," nach ",[573,1415,1417,1432],{"className":1416},[580],[573,1418,1420],{"className":1419},[584],[586,1421,1422],{"xmlns":588},[591,1423,1424,1429],{},[594,1425,1426],{},[604,1427,1428],{},"−",[632,1430,1431],{"encoding":634},"-",[573,1433,1435],{"className":1434,"ariaHidden":640},[639],[573,1436,1438,1441],{"className":1437},[644],[573,1439],{"className":1440,"style":1409},[648],[573,1442,1428],{"className":1443},[653]," ein Hochpunkt, von ",[573,1446,1448,1461],{"className":1447},[580],[573,1449,1451],{"className":1450},[584],[586,1452,1453],{"xmlns":588},[591,1454,1455,1459],{},[594,1456,1457],{},[604,1458,1428],{},[632,1460,1431],{"encoding":634},[573,1462,1464],{"className":1463,"ariaHidden":640},[639],[573,1465,1467,1470],{"className":1466},[644],[573,1468],{"className":1469,"style":1409},[648],[573,1471,1428],{"className":1472},[653],[573,1474,1476,1489],{"className":1475},[580],[573,1477,1479],{"className":1478},[584],[586,1480,1481],{"xmlns":588},[591,1482,1483,1487],{},[594,1484,1485],{},[604,1486,1397],{},[632,1488,1397],{"encoding":634},[573,1490,1492],{"className":1491,"ariaHidden":640},[639],[573,1493,1495,1498],{"className":1494},[644],[573,1496],{"className":1497,"style":1409},[648],[573,1499,1397],{"className":1500},[653]," ein Tiefpunkt, ohne Wechsel kein Extremum.",[565,1503,1504,1505,1571,1572,1671,1672,1700,1701,1804],{},"Die Bedingungen mit ",[573,1506,1508,1530],{"className":1507},[580],[573,1509,1511],{"className":1510},[584],[586,1512,1513],{"xmlns":588},[591,1514,1515,1527],{},[594,1516,1517],{},[597,1518,1519,1521],{},[600,1520,602],{},[594,1522,1523,1525],{},[604,1524,608],{"mathvariant":606},[604,1526,608],{"mathvariant":606},[632,1528,1529],{"encoding":634},"f''",[573,1531,1533],{"className":1532,"ariaHidden":640},[639],[573,1534,1536,1539],{"className":1535},[644],[573,1537],{"className":1538,"style":1277},[648],[573,1540,1542,1545],{"className":1541},[653],[573,1543,602],{"className":1544,"style":658},[653,657],[573,1546,1548],{"className":1547},[662],[573,1549,1551],{"className":1550},[666],[573,1552,1554],{"className":1553},[670],[573,1555,1557],{"className":1556,"style":858},[674],[573,1558,1559,1562],{"style":861},[573,1560],{"className":1561,"style":683},[682],[573,1563,1565],{"className":1564},[687,688,689,690],[573,1566,1568],{"className":1567},[653,690],[573,1569,874],{"className":1570},[653,690]," sind hinreichend, aber nicht notwendig: Bei ",[573,1573,1575,1604],{"className":1574},[580],[573,1576,1578],{"className":1577},[584],[586,1579,1580],{"xmlns":588},[591,1581,1582,1601],{},[594,1583,1584,1586,1588,1590,1592,1594],{},[600,1585,602],{},[604,1587,612],{"stretchy":611},[600,1589,618],{},[604,1591,625],{"stretchy":611},[604,1593,628],{},[597,1595,1596,1598],{},[600,1597,618],{},[620,1599,1600],{},"4",[632,1602,1603],{"encoding":634},"f(x) = x^4",[573,1605,1607,1635],{"className":1606,"ariaHidden":640},[639],[573,1608,1610,1614,1617,1620,1623,1626,1629,1632],{"className":1609},[644],[573,1611],{"className":1612,"style":1613},[648],"height:1em;vertical-align:-0.25em;",[573,1615,602],{"className":1616,"style":658},[653,657],[573,1618,612],{"className":1619},[700],[573,1621,618],{"className":1622},[653,657],[573,1624,625],{"className":1625},[750],[573,1627],{"className":1628,"style":755},[754],[573,1630,628],{"className":1631},[759],[573,1633],{"className":1634,"style":755},[754],[573,1636,1638,1642],{"className":1637},[644],[573,1639],{"className":1640,"style":1641},[648],"height:0.8141em;",[573,1643,1645,1648],{"className":1644},[653],[573,1646,618],{"className":1647},[653,657],[573,1649,1651],{"className":1650},[662],[573,1652,1654],{"className":1653},[666],[573,1655,1657],{"className":1656},[670],[573,1658,1660],{"className":1659,"style":1641},[674],[573,1661,1662,1665],{"style":861},[573,1663],{"className":1664,"style":683},[682],[573,1666,1668],{"className":1667},[687,688,689,690],[573,1669,1600],{"className":1670},[653,690]," gibt es an der Stelle ",[573,1673,1675,1688],{"className":1674},[580],[573,1676,1678],{"className":1677},[584],[586,1679,1680],{"xmlns":588},[591,1681,1682,1686],{},[594,1683,1684],{},[620,1685,622],{},[632,1687,622],{"encoding":634},[573,1689,1691],{"className":1690,"ariaHidden":640},[639],[573,1692,1694,1697],{"className":1693},[644],[573,1695],{"className":1696,"style":769},[648],[573,1698,622],{"className":1699},[653]," ein Minimum, obwohl ",[573,1702,1704,1736],{"className":1703},[580],[573,1705,1707],{"className":1706},[584],[586,1708,1709],{"xmlns":588},[591,1710,1711,1733],{},[594,1712,1713,1723,1725,1727,1729,1731],{},[597,1714,1715,1717],{},[600,1716,602],{},[594,1718,1719,1721],{},[604,1720,608],{"mathvariant":606},[604,1722,608],{"mathvariant":606},[604,1724,612],{"stretchy":611},[620,1726,622],{},[604,1728,625],{"stretchy":611},[604,1730,628],{},[620,1732,622],{},[632,1734,1735],{"encoding":634},"f''(0) = 0",[573,1737,1739,1795],{"className":1738,"ariaHidden":640},[639],[573,1740,1742,1745,1777,1780,1783,1786,1789,1792],{"className":1741},[644],[573,1743],{"className":1744,"style":839},[648],[573,1746,1748,1751],{"className":1747},[653],[573,1749,602],{"className":1750,"style":658},[653,657],[573,1752,1754],{"className":1753},[662],[573,1755,1757],{"className":1756},[666],[573,1758,1760],{"className":1759},[670],[573,1761,1763],{"className":1762,"style":858},[674],[573,1764,1765,1768],{"style":861},[573,1766],{"className":1767,"style":683},[682],[573,1769,1771],{"className":1770},[687,688,689,690],[573,1772,1774],{"className":1773},[653,690],[573,1775,874],{"className":1776},[653,690],[573,1778,612],{"className":1779},[700],[573,1781,622],{"className":1782},[653],[573,1784,625],{"className":1785},[750],[573,1787],{"className":1788,"style":755},[754],[573,1790,628],{"className":1791},[759],[573,1793],{"className":1794,"style":755},[754],[573,1796,1798,1801],{"className":1797},[644],[573,1799],{"className":1800,"style":769},[648],[573,1802,622],{"className":1803},[653]," ist.",[565,1806,1807,1808,1811],{},"Die so gefundenen Extrema sind ",[569,1809,1810],{},"lokal",". Für das globale Maximum oder Minimum vergleicht man sie noch mit den Funktionswerten an den Rändern des Definitionsbereichs.",[560,1813,1815],{"id":1814},"beispiele","Beispiele",[565,1817,1818,1945,1946,2103,2104,2213],{},[573,1819,1821,1856],{"className":1820},[580],[573,1822,1824],{"className":1823},[584],[586,1825,1826],{"xmlns":588},[591,1827,1828,1853],{},[594,1829,1830,1832,1834,1836,1838,1840,1847,1849,1851],{},[600,1831,602],{},[604,1833,612],{"stretchy":611},[600,1835,618],{},[604,1837,625],{"stretchy":611},[604,1839,628],{},[597,1841,1842,1844],{},[600,1843,618],{},[620,1845,1846],{},"3",[604,1848,1428],{},[620,1850,1846],{},[600,1852,618],{},[632,1854,1855],{"encoding":634},"f(x) = x^3 - 3x",[573,1857,1859,1886,1933],{"className":1858,"ariaHidden":640},[639],[573,1860,1862,1865,1868,1871,1874,1877,1880,1883],{"className":1861},[644],[573,1863],{"className":1864,"style":1613},[648],[573,1866,602],{"className":1867,"style":658},[653,657],[573,1869,612],{"className":1870},[700],[573,1872,618],{"className":1873},[653,657],[573,1875,625],{"className":1876},[750],[573,1878],{"className":1879,"style":755},[754],[573,1881,628],{"className":1882},[759],[573,1884],{"className":1885,"style":755},[754],[573,1887,1889,1893,1922,1926,1930],{"className":1888},[644],[573,1890],{"className":1891,"style":1892},[648],"height:0.8974em;vertical-align:-0.0833em;",[573,1894,1896,1899],{"className":1895},[653],[573,1897,618],{"className":1898},[653,657],[573,1900,1902],{"className":1901},[662],[573,1903,1905],{"className":1904},[666],[573,1906,1908],{"className":1907},[670],[573,1909,1911],{"className":1910,"style":1641},[674],[573,1912,1913,1916],{"style":861},[573,1914],{"className":1915,"style":683},[682],[573,1917,1919],{"className":1918},[687,688,689,690],[573,1920,1846],{"className":1921},[653,690],[573,1923],{"className":1924,"style":1925},[754],"margin-right:0.2222em;",[573,1927,1428],{"className":1928},[1929],"mbin",[573,1931],{"className":1932,"style":1925},[754],[573,1934,1936,1939,1942],{"className":1935},[644],[573,1937],{"className":1938,"style":769},[648],[573,1940,1846],{"className":1941},[653],[573,1943,618],{"className":1944},[653,657]," mit ",[573,1947,1949,1988],{"className":1948},[580],[573,1950,1952],{"className":1951},[584],[586,1953,1954],{"xmlns":588},[591,1955,1956,1985],{},[594,1957,1958,1964,1966,1968,1970,1972,1974,1981,1983],{},[597,1959,1960,1962],{},[600,1961,602],{},[604,1963,608],{"mathvariant":606,"lspace":607,"rspace":607},[604,1965,612],{"stretchy":611},[600,1967,618],{},[604,1969,625],{"stretchy":611},[604,1971,628],{},[620,1973,1846],{},[597,1975,1976,1978],{},[600,1977,618],{},[620,1979,1980],{},"2",[604,1982,1428],{},[620,1984,1846],{},[632,1986,1987],{"encoding":634},"f'(x) = 3x^2 - 3",[573,1989,1991,2047,2094],{"className":1990,"ariaHidden":640},[639],[573,1992,1994,1997,2029,2032,2035,2038,2041,2044],{"className":1993},[644],[573,1995],{"className":1996,"style":839},[648],[573,1998,2000,2003],{"className":1999},[653],[573,2001,602],{"className":2002,"style":658},[653,657],[573,2004,2006],{"className":2005},[662],[573,2007,2009],{"className":2008},[666],[573,2010,2012],{"className":2011},[670],[573,2013,2015],{"className":2014,"style":858},[674],[573,2016,2017,2020],{"style":861},[573,2018],{"className":2019,"style":683},[682],[573,2021,2023],{"className":2022},[687,688,689,690],[573,2024,2026],{"className":2025},[653,690],[573,2027,608],{"className":2028},[653,690],[573,2030,612],{"className":2031},[700],[573,2033,618],{"className":2034},[653,657],[573,2036,625],{"className":2037},[750],[573,2039],{"className":2040,"style":755},[754],[573,2042,628],{"className":2043},[759],[573,2045],{"className":2046,"style":755},[754],[573,2048,2050,2053,2056,2085,2088,2091],{"className":2049},[644],[573,2051],{"className":2052,"style":1892},[648],[573,2054,1846],{"className":2055},[653],[573,2057,2059,2062],{"className":2058},[653],[573,2060,618],{"className":2061},[653,657],[573,2063,2065],{"className":2064},[662],[573,2066,2068],{"className":2067},[666],[573,2069,2071],{"className":2070},[670],[573,2072,2074],{"className":2073,"style":1641},[674],[573,2075,2076,2079],{"style":861},[573,2077],{"className":2078,"style":683},[682],[573,2080,2082],{"className":2081},[687,688,689,690],[573,2083,1980],{"className":2084},[653,690],[573,2086],{"className":2087,"style":1925},[754],[573,2089,1428],{"className":2090},[1929],[573,2092],{"className":2093,"style":1925},[754],[573,2095,2097,2100],{"className":2096},[644],[573,2098],{"className":2099,"style":769},[648],[573,2101,1846],{"className":2102},[653]," und ",[573,2105,2107,2142],{"className":2106},[580],[573,2108,2110],{"className":2109},[584],[586,2111,2112],{"xmlns":588},[591,2113,2114,2139],{},[594,2115,2116,2126,2128,2130,2132,2134,2137],{},[597,2117,2118,2120],{},[600,2119,602],{},[594,2121,2122,2124],{},[604,2123,608],{"mathvariant":606},[604,2125,608],{"mathvariant":606},[604,2127,612],{"stretchy":611},[600,2129,618],{},[604,2131,625],{"stretchy":611},[604,2133,628],{},[620,2135,2136],{},"6",[600,2138,618],{},[632,2140,2141],{"encoding":634},"f''(x) = 6x",[573,2143,2145,2201],{"className":2144,"ariaHidden":640},[639],[573,2146,2148,2151,2183,2186,2189,2192,2195,2198],{"className":2147},[644],[573,2149],{"className":2150,"style":839},[648],[573,2152,2154,2157],{"className":2153},[653],[573,2155,602],{"className":2156,"style":658},[653,657],[573,2158,2160],{"className":2159},[662],[573,2161,2163],{"className":2162},[666],[573,2164,2166],{"className":2165},[670],[573,2167,2169],{"className":2168,"style":858},[674],[573,2170,2171,2174],{"style":861},[573,2172],{"className":2173,"style":683},[682],[573,2175,2177],{"className":2176},[687,688,689,690],[573,2178,2180],{"className":2179},[653,690],[573,2181,874],{"className":2182},[653,690],[573,2184,612],{"className":2185},[700],[573,2187,618],{"className":2188},[653,657],[573,2190,625],{"className":2191},[750],[573,2193],{"className":2194,"style":755},[754],[573,2196,628],{"className":2197},[759],[573,2199],{"className":2200,"style":755},[754],[573,2202,2204,2207,2210],{"className":2203},[644],[573,2205],{"className":2206,"style":769},[648],[573,2208,2136],{"className":2209},[653],[573,2211,618],{"className":2212},[653,657],".",[565,2215,2216,2217,2325,2326,2103,2385,2213],{},"Notwendige 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ergibt ",[573,2327,2329,2350],{"className":2328},[580],[573,2330,2332],{"className":2331},[584],[586,2333,2334],{"xmlns":588},[591,2335,2336,2347],{},[594,2337,2338,2340,2342,2344],{},[600,2339,618],{},[604,2341,628],{},[604,2343,1428],{},[620,2345,2346],{},"1",[632,2348,2349],{"encoding":634},"x = 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= -6 \u003C 0 \\;\\Rightarrow\\; \\text{Hochpunkt } H(-1 \\mid 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Die zweite Ableitung liefert keine Entscheidung. ",[573,3215,3217,3251],{"className":3216},[580],[573,3218,3220],{"className":3219},[584],[586,3221,3222],{"xmlns":588},[591,3223,3224,3248],{},[594,3225,3226,3232,3234,3236,3238,3240,3242],{},[597,3227,3228,3230],{},[600,3229,602],{},[604,3231,608],{"mathvariant":606,"lspace":607,"rspace":607},[604,3233,612],{"stretchy":611},[600,3235,618],{},[604,3237,625],{"stretchy":611},[604,3239,628],{},[620,3241,1600],{},[597,3243,3244,3246],{},[600,3245,618],{},[620,3247,1846],{},[632,3249,3250],{"encoding":634},"f'(x) = 4x^3",[573,3252,3254,3310],{"className":3253,"ariaHidden":640},[639],[573,3255,3257,3260,3292,3295,3298,3301,3304,3307],{"className":3256},[644],[573,3258],{"className":3259,"style":839},[648],[573,3261,3263,3266],{"className":3262},[653],[573,3264,602],{"className":3265,"style":658},[653,657],[573,3267,3269],{"className":3268},[662],[573,3270,3272],{"className":3271},[666],[573,3273,3275],{"className":3274},[670],[573,3276,3278],{"className":3277,"style":858},[674],[573,3279,3280,3283],{"style":861},[573,3281],{"className":3282,"style":683},[682],[573,3284,3286],{"className":3285},[687,688,689,690],[573,3287,3289],{"className":3288},[653,690],[573,3290,608],{"className":3291},[653,690],[573,3293,612],{"className":3294},[700],[573,3296,618],{"className":3297},[653,657],[573,3299,625],{"className":3300},[750],[573,3302],{"className":3303,"style":755},[754],[573,3305,628],{"className":3306},[759],[573,3308],{"className":3309,"style":755},[754],[573,3311,3313,3316,3319],{"className":3312},[644],[573,3314],{"className":3315,"style":1641},[648],[573,3317,1600],{"className":3318},[653],[573,3320,3322,3325],{"className":3321},[653],[573,3323,618],{"className":3324},[653,657],[573,3326,3328],{"className":3327},[662],[573,3329,3331],{"className":3330},[666],[573,3332,3334],{"className":3333},[670],[573,3335,3337],{"className":3336,"style":1641},[674],[573,3338,3339,3342],{"style":861},[573,3340],{"className":3341,"style":683},[682],[573,3343,3345],{"className":3344},[687,688,689,690],[573,3346,1846],{"className":3347},[653,690]," wechselt bei ",[573,3350,3352,3365],{"className":3351},[580],[573,3353,3355],{"className":3354},[584],[586,3356,3357],{"xmlns":588},[591,3358,3359,3363],{},[594,3360,3361],{},[620,3362,622],{},[632,3364,622],{"encoding":634},[573,3366,3368],{"className":3367,"ariaHidden":640},[639],[573,3369,3371,3374],{"className":3370},[644],[573,3372],{"className":3373,"style":769},[648],[573,3375,622],{"className":3376},[653]," aber von ",[573,3379,3381,3394],{"className":3380},[580],[573,3382,3384],{"className":3383},[584],[586,3385,3386],{"xmlns":588},[591,3387,3388,3392],{},[594,3389,3390],{},[604,3391,1428],{},[632,3393,1431],{"encoding":634},[573,3395,3397],{"className":3396,"ariaHidden":640},[639],[573,3398,3400,3403],{"className":3399},[644],[573,3401],{"className":3402,"style":1409},[648],[573,3404,1428],{"className":3405},[653],[573,3407,3409,3422],{"className":3408},[580],[573,3410,3412],{"className":3411},[584],[586,3413,3414],{"xmlns":588},[591,3415,3416,3420],{},[594,3417,3418],{},[604,3419,1397],{},[632,3421,1397],{"encoding":634},[573,3423,3425],{"className":3424,"ariaHidden":640},[639],[573,3426,3428,3431],{"className":3427},[644],[573,3429],{"className":3430,"style":1409},[648],[573,3432,1397],{"className":3433},[653],", also ist ",[573,3436,3438,3460],{"className":3437},[580],[573,3439,3441],{"className":3440},[584],[586,3442,3443],{"xmlns":588},[591,3444,3445,3457],{},[594,3446,3447,3449,3451,3453,3455],{},[604,3448,612],{"stretchy":611},[620,3450,622],{},[604,3452,2537],{},[620,3454,622],{},[604,3456,625],{"stretchy":611},[632,3458,3459],{"encoding":634},"(0 \\mid 0)",[573,3461,3463,3484],{"className":3462,"ariaHidden":640},[639],[573,3464,3466,3469,3472,3475,3478,3481],{"className":3465},[644],[573,3467],{"className":3468,"style":1613},[648],[573,3470,612],{"className":3471},[700],[573,3473,622],{"className":3474},[653],[573,3476],{"className":3477,"style":755},[754],[573,3479,2537],{"className":3480},[759],[573,3482],{"className":3483,"style":755},[754],[573,3485,3487,3490,3493],{"className":3486},[644],[573,3488],{"className":3489,"style":1613},[648],[573,3491,622],{"className":3492},[653],[573,3494,625],{"className":3495},[750]," ein Tiefpunkt.",[560,3498,3500],{"id":3499},"typische-fehler","Typische Fehler",[785,3502,3503,3902,4136,4139],{},[788,3504,3505,3506,3645,3646,3011,3742,3840,3841,3901],{},"Nur ",[573,3507,3509,3540],{"className":3508},[580],[573,3510,3512],{"className":3511},[584],[586,3513,3514],{"xmlns":588},[591,3515,3516,3538],{},[594,3517,3518,3524,3526,3532,3534,3536],{},[597,3519,3520,3522],{},[600,3521,602],{},[604,3523,608],{"mathvariant":606,"lspace":607,"rspace":607},[604,3525,612],{"stretchy":611},[614,3527,3528,3530],{},[600,3529,618],{},[620,3531,622],{},[604,3533,625],{"stretchy":611},[604,3535,628],{},[620,3537,622],{},[632,3539,635],{"encoding":634},[573,3541,3543,3636],{"className":3542,"ariaHidden":640},[639],[573,3544,3546,3549,3581,3584,3624,3627,3630,3633],{"className":3545},[644],[573,3547],{"className":3548,"style":839},[648],[573,3550,3552,3555],{"className":3551},[653],[573,3553,602],{"className":3554,"style":658},[653,657],[573,3556,3558],{"className":3557},[662],[573,3559,3561],{"className":3560},[666],[573,3562,3564],{"className":3563},[670],[573,3565,3567],{"className":3566,"style":858},[674],[573,3568,3569,3572],{"style":861},[573,3570],{"className":3571,"style":683},[682],[573,3573,3575],{"className":3574},[687,688,689,690],[573,3576,3578],{"className":3577},[653,690],[573,3579,608],{"className":3580},[653,690],[573,3582,612],{"className":3583},[700],[573,3585,3587,3590],{"className":3586},[653],[573,3588,618],{"className":3589},[653,657],[573,3591,3593],{"className":3592},[662],[573,3594,3596,3616],{"className":3595},[666,713],[573,3597,3599,3613],{"className":3598},[670],[573,3600,3602],{"className":3601,"style":720},[674],[573,3603,3604,3607],{"style":723},[573,3605],{"className":3606,"style":683},[682],[573,3608,3610],{"className":3609},[687,688,689,690],[573,3611,622],{"className":3612},[653,690],[573,3614,737],{"className":3615},[736],[573,3617,3619],{"className":3618},[670],[573,3620,3622],{"className":3621,"style":744},[674],[573,3623],{},[573,3625,625],{"className":3626},[750],[573,3628],{"className":3629,"style":755},[754],[573,3631,628],{"className":3632},[759],[573,3634],{"className":3635,"style":755},[754],[573,3637,3639,3642],{"className":3638},[644],[573,3640],{"className":3641,"style":769},[648],[573,3643,622],{"className":3644},[653]," prüfen. Bei ",[573,3647,3649,3677],{"className":3648},[580],[573,3650,3652],{"className":3651},[584],[586,3653,3654],{"xmlns":588},[591,3655,3656,3674],{},[594,3657,3658,3660,3662,3664,3666,3668],{},[600,3659,602],{},[604,3661,612],{"stretchy":611},[600,3663,618],{},[604,3665,625],{"stretchy":611},[604,3667,628],{},[597,3669,3670,3672],{},[600,3671,618],{},[620,3673,1846],{},[632,3675,3676],{"encoding":634},"f(x) = x^3",[573,3678,3680,3707],{"className":3679,"ariaHidden":640},[639],[573,3681,3683,3686,3689,3692,3695,3698,3701,3704],{"className":3682},[644],[573,3684],{"className":3685,"style":1613},[648],[573,3687,602],{"className":3688,"style":658},[653,657],[573,3690,612],{"className":3691},[700],[573,3693,618],{"className":3694},[653,657],[573,3696,625],{"className":3697},[750],[573,3699],{"className":3700,"style":755},[754],[573,3702,628],{"className":3703},[759],[573,3705],{"className":3706,"style":755},[754],[573,3708,3710,3713],{"className":3709},[644],[573,3711],{"className":3712,"style":1641},[648],[573,3714,3716,3719],{"className":3715},[653],[573,3717,618],{"className":3718},[653,657],[573,3720,3722],{"className":3721},[662],[573,3723,3725],{"className":3724},[666],[573,3726,3728],{"className":3727},[670],[573,3729,3731],{"className":3730,"style":1641},[674],[573,3732,3733,3736],{"style":861},[573,3734],{"className":3735,"style":683},[682],[573,3737,3739],{"className":3738},[687,688,689,690],[573,3740,1846],{"className":3741},[653,690],[573,3743,3745,3772],{"className":3744},[580],[573,3746,3748],{"className":3747},[584],[586,3749,3750],{"xmlns":588},[591,3751,3752,3770],{},[594,3753,3754,3760,3762,3764,3766,3768],{},[597,3755,3756,3758],{},[600,3757,602],{},[604,3759,608],{"mathvariant":606,"lspace":607,"rspace":607},[604,3761,612],{"stretchy":611},[620,3763,622],{},[604,3765,625],{"stretchy":611},[604,3767,628],{},[620,3769,622],{},[632,3771,3042],{"encoding":634},[573,3773,3775,3831],{"className":3774,"ariaHidden":640},[639],[573,3776,3778,3781,3813,3816,3819,3822,3825,3828],{"className":3777},[644],[573,3779],{"className":3780,"style":839},[648],[573,3782,3784,3787],{"className":3783},[653],[573,3785,602],{"className":3786,"style":658},[653,657],[573,3788,3790],{"className":3789},[662],[573,3791,3793],{"className":3792},[666],[573,3794,3796],{"className":3795},[670],[573,3797,3799],{"className":3798,"style":858},[674],[573,3800,3801,3804],{"style":861},[573,3802],{"className":3803,"style":683},[682],[573,3805,3807],{"className":3806},[687,688,689,690],[573,3808,3810],{"className":3809},[653,690],[573,3811,608],{"className":3812},[653,690],[573,3814,612],{"className":3815},[700],[573,3817,622],{"className":3818},[653],[573,3820,625],{"className":3821},[750],[573,3823],{"className":3824,"style":755},[754],[573,3826,628],{"className":3827},[759],[573,3829],{"className":3830,"style":755},[754],[573,3832,3834,3837],{"className":3833},[644],[573,3835],{"className":3836,"style":769},[648],[573,3838,622],{"className":3839},[653],", aber ",[573,3842,3844,3865],{"className":3843},[580],[573,3845,3847],{"className":3846},[584],[586,3848,3849],{"xmlns":588},[591,3850,3851,3863],{},[594,3852,3853,3855,3857,3859,3861],{},[604,3854,612],{"stretchy":611},[620,3856,622],{},[604,3858,2537],{},[620,3860,622],{},[604,3862,625],{"stretchy":611},[632,3864,3459],{"encoding":634},[573,3866,3868,3889],{"className":3867,"ariaHidden":640},[639],[573,3869,3871,3874,3877,3880,3883,3886],{"className":3870},[644],[573,3872],{"className":3873,"style":1613},[648],[573,3875,612],{"className":3876},[700],[573,3878,622],{"className":3879},[653],[573,3881],{"className":3882,"style":755},[754],[573,3884,2537],{"className":3885},[759],[573,3887],{"className":3888,"style":755},[754],[573,3890,3892,3895,3898],{"className":3891},[644],[573,3893],{"className":3894,"style":1613},[648],[573,3896,622],{"className":3897},[653],[573,3899,625],{"className":3900},[750]," ist ein Sattelpunkt.",[788,3903,3904,3905,4048,4049,1310,4108,2213],{},"Aus ",[573,3906,3908,3943],{"className":3907},[580],[573,3909,3911],{"className":3910},[584],[586,3912,3913],{"xmlns":588},[591,3914,3915,3941],{},[594,3916,3917,3927,3929,3935,3937,3939],{},[597,3918,3919,3921],{},[600,3920,602],{},[594,3922,3923,3925],{},[604,3924,608],{"mathvariant":606},[604,3926,608],{"mathvariant":606},[604,3928,612],{"stretchy":611},[614,3930,3931,3933],{},[600,3932,618],{},[620,3934,622],{},[604,3936,625],{"stretchy":611},[604,3938,628],{},[620,3940,622],{},[632,3942,1136],{"encoding":634},[573,3944,3946,4039],{"className":3945,"ariaHidden":640},[639],[573,3947,3949,3952,3984,3987,4027,4030,4033,4036],{"className":3948},[644],[573,3950],{"className":3951,"style":839},[648],[573,3953,3955,3958],{"className":3954},[653],[573,3956,602],{"className":3957,"style":658},[653,657],[573,3959,3961],{"className":3960},[662],[573,3962,3964],{"className":3963},[666],[573,3965,3967],{"className":3966},[670],[573,3968,3970],{"className":3969,"style":858},[674],[573,3971,3972,3975],{"style":861},[573,3973],{"className":3974,"style":683},[682],[573,3976,3978],{"className":3977},[687,688,689,690],[573,3979,3981],{"className":3980},[653,690],[573,3982,874],{"className":3983},[653,690],[573,3985,612],{"className":3986},[700],[573,3988,3990,3993],{"className":3989},[653],[573,3991,618],{"className":3992},[653,657],[573,3994,3996],{"className":3995},[662],[573,3997,3999,4019],{"className":3998},[666,713],[573,4000,4002,4016],{"className":4001},[670],[573,4003,4005],{"className":4004,"style":720},[674],[573,4006,4007,4010],{"style":723},[573,4008],{"className":4009,"style":683},[682],[573,4011,4013],{"className":4012},[687,688,689,690],[573,4014,622],{"className":4015},[653,690],[573,4017,737],{"className":4018},[736],[573,4020,4022],{"className":4021},[670],[573,4023,4025],{"className":4024,"style":744},[674],[573,4026],{},[573,4028,625],{"className":4029},[750],[573,4031],{"className":4032,"style":755},[754],[573,4034,628],{"className":4035},[759],[573,4037],{"className":4038,"style":755},[754],[573,4040,4042,4045],{"className":4041},[644],[573,4043],{"className":4044,"style":769},[648],[573,4046,622],{"className":4047},[653]," schließen, dass kein Extremum vorliegt. Gegenbeispiel: ",[573,4050,4052,4070],{"className":4051},[580],[573,4053,4055],{"className":4054},[584],[586,4056,4057],{"xmlns":588},[591,4058,4059,4067],{},[594,4060,4061],{},[597,4062,4063,4065],{},[600,4064,618],{},[620,4066,1600],{},[632,4068,4069],{"encoding":634},"x^4",[573,4071,4073],{"className":4072,"ariaHidden":640},[639],[573,4074,4076,4079],{"className":4075},[644],[573,4077],{"className":4078,"style":1641},[648],[573,4080,4082,4085],{"className":4081},[653],[573,4083,618],{"className":4084},[653,657],[573,4086,4088],{"className":4087},[662],[573,4089,4091],{"className":4090},[666],[573,4092,4094],{"className":4093},[670],[573,4095,4097],{"className":4096,"style":1641},[674],[573,4098,4099,4102],{"style":861},[573,4100],{"className":4101,"style":683},[682],[573,4103,4105],{"className":4104},[687,688,689,690],[573,4106,1600],{"className":4107},[653,690],[573,4109,4111,4124],{"className":4110},[580],[573,4112,4114],{"className":4113},[584],[586,4115,4116],{"xmlns":588},[591,4117,4118,4122],{},[594,4119,4120],{},[620,4121,622],{},[632,4123,622],{"encoding":634},[573,4125,4127],{"className":4126,"ariaHidden":640},[639],[573,4128,4130,4133],{"className":4129},[644],[573,4131],{"className":4132,"style":769},[648],[573,4134,622],{"className":4135},[653],[788,4137,4138],{},"Hoch- und Tiefpunkt verwechseln. Negative zweite Ableitung bedeutet Hochpunkt.",[788,4140,4141,4142,4170,4171,4199,4200,4229,4230,2213],{},"Den ",[573,4143,4145,4158],{"className":4144},[580],[573,4146,4148],{"className":4147},[584],[586,4149,4150],{"xmlns":588},[591,4151,4152,4156],{},[594,4153,4154],{},[600,4155,2453],{},[632,4157,2453],{"encoding":634},[573,4159,4161],{"className":4160,"ariaHidden":640},[639],[573,4162,4164,4167],{"className":4163},[644],[573,4165],{"className":4166,"style":2465},[648],[573,4168,2453],{"className":4169,"style":2469},[653,657],"-Wert vergessen. Ein Extrempunkt braucht beide Koordinaten, der ",[573,4172,4174,4187],{"className":4173},[580],[573,4175,4177],{"className":4176},[584],[586,4178,4179],{"xmlns":588},[591,4180,4181,4185],{},[594,4182,4183],{},[600,4184,2453],{},[632,4186,2453],{"encoding":634},[573,4188,4190],{"className":4189,"ariaHidden":640},[639],[573,4191,4193,4196],{"className":4192},[644],[573,4194],{"className":4195,"style":2465},[648],[573,4197,2453],{"className":4198,"style":2469},[653,657],"-Wert kommt aus ",[573,4201,4203,4216],{"className":4202},[580],[573,4204,4206],{"className":4205},[584],[586,4207,4208],{"xmlns":588},[591,4209,4210,4214],{},[594,4211,4212],{},[600,4213,602],{},[632,4215,602],{"encoding":634},[573,4217,4219],{"className":4218,"ariaHidden":640},[639],[573,4220,4222,4226],{"className":4221},[644],[573,4223],{"className":4224,"style":4225},[648],"height:0.8889em;vertical-align:-0.1944em;",[573,4227,602],{"className":4228,"style":658},[653,657],", nicht aus 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encoding=\"application\u002Fx-tex\">\\begin{aligned} f'(x_0) = 0 \\;\\wedge\\; f''(x_0) < 0 &\\;\\Rightarrow\\; \\text{Hochpunkt} \\\\ f'(x_0) = 0 \\;\\wedge\\; f''(x_0) > 0 &\\;\\Rightarrow\\; \\text{Tiefpunkt} \\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\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=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">∧\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\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>\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\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=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">∧\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\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>\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\">Hochpunkt\u003C\u002Fspan>\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\">\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\">Tiefpunkt\u003C\u002Fspan>\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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},"\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbedingungen-extremstellen",[],[4383,4384,4385],"inflection-condition","power-rule","tangent-line",{"title":548,"description":4292},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbedingungen-extremstellen","Ist die erste Ableitung null und die zweite ungleich null, liegt ein Hoch- oder Tiefpunkt vor.",[4390,4391,4392,4393,4394],"extremstelle","hochpunkt","tiefpunkt","kurvendiskussion","ableitung",[4396,4399,4403],{"symbol":1331,"meaning":4397,"symbolHtml":4398},"untersuchte Stelle","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x_0\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:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4400,"meaning":4401,"symbolHtml":4402},"f'(x_0)","erste Ableitung an der Stelle x₀, Steigung des Graphen","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f'(x_0)\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:1.0019em;vertical-align:-0.25em;\">\u003C\u002Fspan>\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:0.7519em;\">\u003Cspan style=\"top:-3.063em;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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;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\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" 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