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f_{xy}^2",[569,1079,1081,1101,1166,1273],{"className":1080,"ariaHidden":601},[617],[569,1082,1084,1088,1092,1095,1098],{"className":1083},[621],[569,1085],{"className":1086,"style":1087},[625],"height:0.6833em;",[569,1089,1020],{"className":1090,"style":1091},[630,631],"margin-right:0.0278em;",[569,1093],{"className":1094,"style":722},[647],[569,1096,689],{"className":1097},[726],[569,1099],{"className":1100,"style":722},[647],[569,1102,1104,1107,1111,1114,1157,1160,1163],{"className":1103},[621],[569,1105],{"className":1106,"style":935},[625],[569,1108,1025],{"className":1109},[1110],"mop",[569,1112],{"className":1113,"style":648},[647],[569,1115,1117,1121],{"className":1116},[630],[569,1118,1033],{"className":1119,"style":1120},[630,631],"margin-right:0.0813em;",[569,1122,1124],{"className":1123},[838],[569,1125,1127,1149],{"className":1126},[744,842],[569,1128,1130,1146],{"className":1129},[748],[569,1131,1134],{"className":1132,"style":1133},[752],"height:0.3361em;",[569,1135,1137,1140],{"style":1136},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[569,1138],{"className":1139,"style":856},[759],[569,1141,1143],{"className":1142},[860,861,862,863],[569,1144,590],{"className":1145,"style":632},[630,631,863],[569,1147,871],{"className":1148},[870],[569,1150,1152],{"className":1151},[748],[569,1153,1155],{"className":1154,"style":974},[752],[569,1156],{},[569,1158],{"className":1159,"style":722},[647],[569,1161,689],{"className":1162},[726],[569,1164],{"className":1165,"style":722},[647],[569,1167,1169,1172,1216,1262,1266,1270],{"className":1168},[621],[569,1170],{"className":1171,"style":935},[625],[569,1173,1175,1178],{"className":1174},[630],[569,1176,590],{"className":1177,"style":632},[630,631],[569,1179,1181],{"className":1180},[838],[569,1182,1184,1208],{"className":1183},[744,842],[569,1185,1187,1205],{"className":1186},[748],[569,1188,1190],{"className":1189,"style":849},[752],[569,1191,1192,1195],{"style":852},[569,1193],{"className":1194,"style":856},[759],[569,1196,1198],{"className":1197},[860,861,862,863],[569,1199,1201],{"className":1200},[630,863],[569,1202,1204],{"className":1203},[630,631,863],"xx",[569,1206,871],{"className":1207},[870],[569,1209,1211],{"className":1210},[748],[569,1212,1214],{"className":1213,"style":878},[752],[569,1215],{},[569,1217,1219,1222],{"className":1218},[630],[569,1220,590],{"className":1221,"style":632},[630,631],[569,1223,1225],{"className":1224},[838],[569,1226,1228,1254],{"className":1227},[744,842],[569,1229,1231,1251],{"className":1230},[748],[569,1232,1234],{"className":1233,"style":849},[752],[569,1235,1236,1239],{"style":852},[569,1237],{"className":1238,"style":856},[759],[569,1240,1242],{"className":1241},[860,861,862,863],[569,1243,1245,1248],{"className":1244},[630,863],[569,1246,605],{"className":1247,"style":652},[630,631,863],[569,1249,605],{"className":1250,"style":652},[630,631,863],[569,1252,871],{"className":1253},[870],[569,1255,1257],{"className":1256},[748],[569,1258,1260],{"className":1259,"style":974},[752],[569,1261],{},[569,1263],{"className":1264,"style":1265},[647],"margin-right:0.2222em;",[569,1267,1060],{"className":1268},[1269],"mbin",[569,1271],{"className":1272,"style":1265},[647],[569,1274,1276,1280],{"className":1275},[621],[569,1277],{"className":1278,"style":1279},[625],"height:1.1972em;vertical-align:-0.3831em;",[569,1281,1283,1286],{"className":1282},[630],[569,1284,590],{"className":1285,"style":632},[630,631],[569,1287,1289],{"className":1288},[838],[569,1290,1292,1332],{"className":1291},[744,842],[569,1293,1295,1329],{"className":1294},[748],[569,1296,1299,1317],{"className":1297,"style":1298},[752],"height:0.8141em;",[569,1300,1302,1305],{"style":1301},"top:-2.453em;margin-left:-0.1076em;margin-right:0.05em;",[569,1303],{"className":1304,"style":856},[759],[569,1306,1308],{"className":1307},[860,861,862,863],[569,1309,1311,1314],{"className":1310},[630,863],[569,1312,598],{"className":1313},[630,631,863],[569,1315,605],{"className":1316,"style":652},[630,631,863],[569,1318,1320,1323],{"style":1319},"top:-3.063em;margin-right:0.05em;",[569,1321],{"className":1322,"style":856},[759],[569,1324,1326],{"className":1325},[860,861,862,863],[569,1327,1074],{"className":1328},[630,863],[569,1330,871],{"className":1331},[870],[569,1333,1335],{"className":1334},[748],[569,1336,1339],{"className":1337,"style":1338},[752],"height:0.3831em;",[569,1340],{},[565,1342,1343,1344,1397,1398,1475,1476,1528,1529,1532,1533,1561,1562,1590,1591,1642,1643,1646],{},"Bei ",[569,1345,1347,1366],{"className":1346},[572],[569,1348,1350],{"className":1349},[576],[578,1351,1352],{"xmlns":580},[582,1353,1354,1363],{},[585,1355,1356,1358,1361],{},[588,1357,1020],{},[592,1359,1360],{},">",[694,1362,696],{},[610,1364,1365],{"encoding":612},"D > 0",[569,1367,1369,1388],{"className":1368,"ariaHidden":601},[617],[569,1370,1372,1376,1379,1382,1385],{"className":1371},[621],[569,1373],{"className":1374,"style":1375},[625],"height:0.7224em;vertical-align:-0.0391em;",[569,1377,1020],{"className":1378,"style":1091},[630,631],[569,1380],{"className":1381,"style":722},[647],[569,1383,1360],{"className":1384},[726],[569,1386],{"className":1387,"style":722},[647],[569,1389,1391,1394],{"className":1390},[621],[569,1392],{"className":1393,"style":896},[625],[569,1395,696],{"className":1396},[630]," liegt ein Extremum vor, und das Vorzeichen von ",[569,1399,1401,1423],{"className":1400},[572],[569,1402,1404],{"className":1403},[576],[578,1405,1406],{"xmlns":580},[582,1407,1408,1420],{},[585,1409,1410],{},[807,1411,1412,1414],{},[588,1413,590],{},[585,1415,1416,1418],{},[588,1417,598],{},[588,1419,598],{},[610,1421,1422],{"encoding":612},"f_{xx}",[569,1424,1426],{"className":1425,"ariaHidden":601},[617],[569,1427,1429,1432],{"className":1428},[621],[569,1430],{"className":1431,"style":712},[625],[569,1433,1435,1438],{"className":1434},[630],[569,1436,590],{"className":1437,"style":632},[630,631],[569,1439,1441],{"className":1440},[838],[569,1442,1444,1467],{"className":1443},[744,842],[569,1445,1447,1464],{"className":1446},[748],[569,1448,1450],{"className":1449,"style":849},[752],[569,1451,1452,1455],{"style":852},[569,1453],{"className":1454,"style":856},[759],[569,1456,1458],{"className":1457},[860,861,862,863],[569,1459,1461],{"className":1460},[630,863],[569,1462,1204],{"className":1463},[630,631,863],[569,1465,871],{"className":1466},[870],[569,1468,1470],{"className":1469},[748],[569,1471,1473],{"className":1472,"style":878},[752],[569,1474],{}," entscheidet über Minimum oder Maximum. Bei ",[569,1477,1479,1498],{"className":1478},[572],[569,1480,1482],{"className":1481},[576],[578,1483,1484],{"xmlns":580},[582,1485,1486,1495],{},[585,1487,1488,1490,1493],{},[588,1489,1020],{},[592,1491,1492],{},"\u003C",[694,1494,696],{},[610,1496,1497],{"encoding":612},"D \u003C 0",[569,1499,1501,1519],{"className":1500,"ariaHidden":601},[617],[569,1502,1504,1507,1510,1513,1516],{"className":1503},[621],[569,1505],{"className":1506,"style":1375},[625],[569,1508,1020],{"className":1509,"style":1091},[630,631],[569,1511],{"className":1512,"style":722},[647],[569,1514,1492],{"className":1515},[726],[569,1517],{"className":1518,"style":722},[647],[569,1520,1522,1525],{"className":1521},[621],[569,1523],{"className":1524,"style":896},[625],[569,1526,696],{"className":1527},[630]," ist der Punkt ein ",[665,1530,1531],{},"Sattelpunkt",": In einer Richtung steigt ",[569,1534,1536,1549],{"className":1535},[572],[569,1537,1539],{"className":1538},[576],[578,1540,1541],{"xmlns":580},[582,1542,1543,1547],{},[585,1544,1545],{},[588,1546,590],{},[610,1548,590],{"encoding":612},[569,1550,1552],{"className":1551,"ariaHidden":601},[617],[569,1553,1555,1558],{"className":1554},[621],[569,1556],{"className":1557,"style":712},[625],[569,1559,590],{"className":1560,"style":632},[630,631],", in einer anderen fällt ",[569,1563,1565,1578],{"className":1564},[572],[569,1566,1568],{"className":1567},[576],[578,1569,1570],{"xmlns":580},[582,1571,1572,1576],{},[585,1573,1574],{},[588,1575,590],{},[610,1577,590],{"encoding":612},[569,1579,1581],{"className":1580,"ariaHidden":601},[617],[569,1582,1584,1587],{"className":1583},[621],[569,1585],{"className":1586,"style":712},[625],[569,1588,590],{"className":1589,"style":632},[630,631],". Bei ",[569,1592,1594,1612],{"className":1593},[572],[569,1595,1597],{"className":1596},[576],[578,1598,1599],{"xmlns":580},[582,1600,1601,1609],{},[585,1602,1603,1605,1607],{},[588,1604,1020],{},[592,1606,689],{},[694,1608,696],{},[610,1610,1611],{"encoding":612},"D = 0",[569,1613,1615,1633],{"className":1614,"ariaHidden":601},[617],[569,1616,1618,1621,1624,1627,1630],{"className":1617},[621],[569,1619],{"className":1620,"style":1087},[625],[569,1622,1020],{"className":1623,"style":1091},[630,631],[569,1625],{"className":1626,"style":722},[647],[569,1628,689],{"className":1629},[726],[569,1631],{"className":1632,"style":722},[647],[569,1634,1636,1639],{"className":1635},[621],[569,1637],{"className":1638,"style":896},[625],[569,1640,696],{"className":1641},[630]," ist ",[665,1644,1645],{},"keine Aussage"," möglich; dann muss man die Funktion in der Umgebung direkt untersuchen.",[565,1648,1649],{},"Das Verfahren findet nur lokale Extrema im Inneren des Definitionsbereichs. Globale Extrema auf einem beschränkten Bereich können auch auf dem Rand liegen; den untersucht man getrennt, z.B. mit Lagrange-Multiplikatoren.",[560,1651,1653],{"id":1652},"beispiele","Beispiele",[565,1655,1656,1657,1642,1856,900,2010,2164,2165,2256,2257,2339,2340,2431,2432,793,2515,2566,2567,2619,2620,900,2672,999],{},"Für ",[569,1658,1660,1711],{"className":1659},[572],[569,1661,1663],{"className":1662},[576],[578,1664,1665],{"xmlns":580},[582,1666,1667,1708],{},[585,1668,1669,1671,1673,1675,1677,1679,1681,1683,1691,1693,1695,1697,1699,1702],{},[588,1670,590],{},[592,1672,595],{"stretchy":594},[588,1674,598],{},[592,1676,602],{"separator":601},[588,1678,605],{},[592,1680,608],{"stretchy":594},[592,1682,689],{},[1684,1685,1686,1688],"msup",{},[588,1687,598],{},[694,1689,1690],{},"3",[592,1692,1060],{},[694,1694,1690],{},[588,1696,598],{},[588,1698,605],{},[592,1700,1701],{},"+",[1684,1703,1704,1706],{},[588,1705,605],{},[694,1707,1690],{},[610,1709,1710],{"encoding":612},"f(x, y) = x^3 - 3xy + y^3",[569,1712,1714,1750,1795,1820],{"className":1713,"ariaHidden":601},[617],[569,1715,1717,1720,1723,1726,1729,1732,1735,1738,1741,1744,1747],{"className":1716},[621],[569,1718],{"className":1719,"style":626},[625],[569,1721,590],{"className":1722,"style":632},[630,631],[569,1724,595],{"className":1725},[636],[569,1727,598],{"className":1728},[630,631],[569,1730,602],{"className":1731},[643],[569,1733],{"className":1734,"style":648},[647],[569,1736,605],{"className":1737,"style":652},[630,631],[569,1739,608],{"className":1740},[656],[569,1742],{"className":1743,"style":722},[647],[569,1745,689],{"className":1746},[726],[569,1748],{"className":1749,"style":722},[647],[569,1751,1753,1757,1786,1789,1792],{"className":1752},[621],[569,1754],{"className":1755,"style":1756},[625],"height:0.8974em;vertical-align:-0.0833em;",[569,1758,1760,1763],{"className":1759},[630],[569,1761,598],{"className":1762},[630,631],[569,1764,1766],{"className":1765},[838],[569,1767,1769],{"className":1768},[744],[569,1770,1772],{"className":1771},[748],[569,1773,1775],{"className":1774,"style":1298},[752],[569,1776,1777,1780],{"style":1319},[569,1778],{"className":1779,"style":856},[759],[569,1781,1783],{"className":1782},[860,861,862,863],[569,1784,1690],{"className":1785},[630,863],[569,1787],{"className":1788,"style":1265},[647],[569,1790,1060],{"className":1791},[1269],[569,1793],{"className":1794,"style":1265},[647],[569,1796,1798,1802,1805,1808,1811,1814,1817],{"className":1797},[621],[569,1799],{"className":1800,"style":1801},[625],"height:0.8389em;vertical-align:-0.1944em;",[569,1803,1690],{"className":1804},[630],[569,1806,598],{"className":1807},[630,631],[569,1809,605],{"className":1810,"style":652},[630,631],[569,1812],{"className":1813,"style":1265},[647],[569,1815,1701],{"className":1816},[1269],[569,1818],{"className":1819,"style":1265},[647],[569,1821,1823,1827],{"className":1822},[621],[569,1824],{"className":1825,"style":1826},[625],"height:1.0085em;vertical-align:-0.1944em;",[569,1828,1830,1833],{"className":1829},[630],[569,1831,605],{"className":1832,"style":652},[630,631],[569,1834,1836],{"className":1835},[838],[569,1837,1839],{"className":1838},[744],[569,1840,1842],{"className":1841},[748],[569,1843,1845],{"className":1844,"style":1298},[752],[569,1846,1847,1850],{"style":1319},[569,1848],{"className":1849,"style":856},[759],[569,1851,1853],{"className":1852},[860,861,862,863],[569,1854,1690],{"className":1855},[630,863],[569,1857,1859,1893],{"className":1858},[572],[569,1860,1862],{"className":1861},[576],[578,1863,1864],{"xmlns":580},[582,1865,1866,1890],{},[585,1867,1868,1874,1876,1878,1884,1886,1888],{},[807,1869,1870,1872],{},[588,1871,590],{},[588,1873,598],{},[592,1875,689],{},[694,1877,1690],{},[1684,1879,1880,1882],{},[588,1881,598],{},[694,1883,1074],{},[592,1885,1060],{},[694,1887,1690],{},[588,1889,605],{},[610,1891,1892],{"encoding":612},"f_x = 3x^2 - 3y",[569,1894,1896,1951,1998],{"className":1895,"ariaHidden":601},[617],[569,1897,1899,1902,1942,1945,1948],{"className":1898},[621],[569,1900],{"className":1901,"style":712},[625],[569,1903,1905,1908],{"className":1904},[630],[569,1906,590],{"className":1907,"style":632},[630,631],[569,1909,1911],{"className":1910},[838],[569,1912,1914,1934],{"className":1913},[744,842],[569,1915,1917,1931],{"className":1916},[748],[569,1918,1920],{"className":1919,"style":849},[752],[569,1921,1922,1925],{"style":852},[569,1923],{"className":1924,"style":856},[759],[569,1926,1928],{"className":1927},[860,861,862,863],[569,1929,598],{"className":1930},[630,631,863],[569,1932,871],{"className":1933},[870],[569,1935,1937],{"className":1936},[748],[569,1938,1940],{"className":1939,"style":878},[752],[569,1941],{},[569,1943],{"className":1944,"style":722},[647],[569,1946,689],{"className":1947},[726],[569,1949],{"className":1950,"style":722},[647],[569,1952,1954,1957,1960,1989,1992,1995],{"className":1953},[621],[569,1955],{"className":1956,"style":1756},[625],[569,1958,1690],{"className":1959},[630],[569,1961,1963,1966],{"className":1962},[630],[569,1964,598],{"className":1965},[630,631],[569,1967,1969],{"className":1968},[838],[569,1970,1972],{"className":1971},[744],[569,1973,1975],{"className":1974},[748],[569,1976,1978],{"className":1977,"style":1298},[752],[569,1979,1980,1983],{"style":1319},[569,1981],{"className":1982,"style":856},[759],[569,1984,1986],{"className":1985},[860,861,862,863],[569,1987,1074],{"className":1988},[630,863],[569,1990],{"className":1991,"style":1265},[647],[569,1993,1060],{"className":1994},[1269],[569,1996],{"className":1997,"style":1265},[647],[569,1999,2001,2004,2007],{"className":2000},[621],[569,2002],{"className":2003,"style":1801},[625],[569,2005,1690],{"className":2006},[630],[569,2008,605],{"className":2009,"style":652},[630,631],[569,2011,2013,2047],{"className":2012},[572],[569,2014,2016],{"className":2015},[576],[578,2017,2018],{"xmlns":580},[582,2019,2020,2044],{},[585,2021,2022,2028,2030,2032,2038,2040,2042],{},[807,2023,2024,2026],{},[588,2025,590],{},[588,2027,605],{},[592,2029,689],{},[694,2031,1690],{},[1684,2033,2034,2036],{},[588,2035,605],{},[694,2037,1074],{},[592,2039,1060],{},[694,2041,1690],{},[588,2043,598],{},[610,2045,2046],{"encoding":612},"f_y = 3y^2 - 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ergibt sich ",[569,2433,2435,2458],{"className":2434},[572],[569,2436,2438],{"className":2437},[576],[578,2439,2440],{"xmlns":580},[582,2441,2442,2455],{},[585,2443,2444,2451,2453],{},[1684,2445,2446,2448],{},[588,2447,598],{},[694,2449,2450],{},"4",[592,2452,689],{},[588,2454,598],{},[610,2456,2457],{"encoding":612},"x^4 = 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Die kritischen Punkte sind ",[569,2621,2623,2645],{"className":2622},[572],[569,2624,2626],{"className":2625},[576],[578,2627,2628],{"xmlns":580},[582,2629,2630,2642],{},[585,2631,2632,2634,2636,2638,2640],{},[592,2633,595],{"stretchy":594},[694,2635,696],{},[592,2637,602],{"separator":601},[694,2639,696],{},[592,2641,608],{"stretchy":594},[610,2643,2644],{"encoding":612},"(0, 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6x",[569,2760,2762,2820],{"className":2761,"ariaHidden":601},[617],[569,2763,2765,2768,2811,2814,2817],{"className":2764},[621],[569,2766],{"className":2767,"style":712},[625],[569,2769,2771,2774],{"className":2770},[630],[569,2772,590],{"className":2773,"style":632},[630,631],[569,2775,2777],{"className":2776},[838],[569,2778,2780,2803],{"className":2779},[744,842],[569,2781,2783,2800],{"className":2782},[748],[569,2784,2786],{"className":2785,"style":849},[752],[569,2787,2788,2791],{"style":852},[569,2789],{"className":2790,"style":856},[759],[569,2792,2794],{"className":2793},[860,861,862,863],[569,2795,2797],{"className":2796},[630,863],[569,2798,1204],{"className":2799},[630,631,863],[569,2801,871],{"className":2802},[870],[569,2804,2806],{"className":2805},[748],[569,2807,2809],{"className":2808,"style":878},[752],[569,2810],{},[569,2812],{"className":2813,"style":722},[647],[569,2815,689],{"className":2816},[726],[569,2818],{"className":2819,"style":722},[647],[569,2821,2823,2826,2829],{"className":2822},[621],[569,2824],{"className":2825,"style":896},[625],[569,2827,2753],{"className":2828},[630],[569,2830,598],{"className":2831},[630,631],", ",[569,2834,2836,2864],{"className":2835},[572],[569,2837,2839],{"className":2838},[576],[578,2840,2841],{"xmlns":580},[582,2842,2843,2861],{},[585,2844,2845,2855,2857,2859],{},[807,2846,2847,2849],{},[588,2848,590],{},[585,2850,2851,2853],{},[588,2852,605],{},[588,2854,605],{},[592,2856,689],{},[694,2858,2753],{},[588,2860,605],{},[610,2862,2863],{"encoding":612},"f_{yy} = 6y",[569,2865,2867,2928],{"className":2866,"ariaHidden":601},[617],[569,2868,2870,2873,2919,2922,2925],{"className":2869},[621],[569,2871],{"className":2872,"style":935},[625],[569,2874,2876,2879],{"className":2875},[630],[569,2877,590],{"className":2878,"style":632},[630,631],[569,2880,2882],{"className":2881},[838],[569,2883,2885,2911],{"className":2884},[744,842],[569,2886,2888,2908],{"className":2887},[748],[569,2889,2891],{"className":2890,"style":849},[752],[569,2892,2893,2896],{"style":852},[569,2894],{"className":2895,"style":856},[759],[569,2897,2899],{"className":2898},[860,861,862,863],[569,2900,2902,2905],{"className":2901},[630,863],[569,2903,605],{"className":2904,"style":652},[630,631,863],[569,2906,605],{"className":2907,"style":652},[630,631,863],[569,2909,871],{"className":2910},[870],[569,2912,2914],{"className":2913},[748],[569,2915,2917],{"className":2916,"style":974},[752],[569,2918],{},[569,2920],{"className":2921,"style":722},[647],[569,2923,689],{"className":2924},[726],[569,2926],{"className":2927,"style":722},[647],[569,2929,2931,2934,2937],{"className":2930},[621],[569,2932],{"className":2933,"style":1801},[625],[569,2935,2753],{"className":2936},[630],[569,2938,605],{"className":2939,"style":652},[630,631],[569,2941,2943,2971],{"className":2942},[572],[569,2944,2946],{"className":2945},[576],[578,2947,2948],{"xmlns":580},[582,2949,2950,2968],{},[585,2951,2952,2962,2964,2966],{},[807,2953,2954,2956],{},[588,2955,590],{},[585,2957,2958,2960],{},[588,2959,598],{},[588,2961,605],{},[592,2963,689],{},[592,2965,1060],{},[694,2967,1690],{},[610,2969,2970],{"encoding":612},"f_{xy} = -3",[569,2972,2974,3035],{"className":2973,"ariaHidden":601},[617],[569,2975,2977,2980,3026,3029,3032],{"className":2976},[621],[569,2978],{"className":2979,"style":935},[625],[569,2981,2983,2986],{"className":2982},[630],[569,2984,590],{"className":2985,"style":632},[630,631],[569,2987,2989],{"className":2988},[838],[569,2990,2992,3018],{"className":2991},[744,842],[569,2993,2995,3015],{"className":2994},[748],[569,2996,2998],{"className":2997,"style":849},[752],[569,2999,3000,3003],{"style":852},[569,3001],{"className":3002,"style":856},[759],[569,3004,3006],{"className":3005},[860,861,862,863],[569,3007,3009,3012],{"className":3008},[630,863],[569,3010,598],{"className":3011},[630,631,863],[569,3013,605],{"className":3014,"style":652},[630,631,863],[569,3016,871],{"className":3017},[870],[569,3019,3021],{"className":3020},[748],[569,3022,3024],{"className":3023,"style":974},[752],[569,3025],{},[569,3027],{"className":3028,"style":722},[647],[569,3030,689],{"className":3031},[726],[569,3033],{"className":3034,"style":722},[647],[569,3036,3038,3042,3045],{"className":3037},[621],[569,3039],{"className":3040,"style":3041},[625],"height:0.7278em;vertical-align:-0.0833em;",[569,3043,1060],{"className":3044},[630],[569,3046,1690],{"className":3047},[630],[569,3049,3051,3079],{"className":3050},[572],[569,3052,3054],{"className":3053},[576],[578,3055,3056],{"xmlns":580},[582,3057,3058,3076],{},[585,3059,3060,3062,3064,3067,3069,3071,3073],{},[588,3061,1020],{},[592,3063,689],{},[694,3065,3066],{},"36",[588,3068,598],{},[588,3070,605],{},[592,3072,1060],{},[694,3074,3075],{},"9",[610,3077,3078],{"encoding":612},"D = 36xy - 9",[569,3080,3082,3100,3124],{"className":3081,"ariaHidden":601},[617],[569,3083,3085,3088,3091,3094,3097],{"className":3084},[621],[569,3086],{"className":3087,"style":1087},[625],[569,3089,1020],{"className":3090,"style":1091},[630,631],[569,3092],{"className":3093,"style":722},[647],[569,3095,689],{"className":3096},[726],[569,3098],{"className":3099,"style":722},[647],[569,3101,3103,3106,3109,3112,3115,3118,3121],{"className":3102},[621],[569,3104],{"className":3105,"style":1801},[625],[569,3107,3066],{"className":3108},[630],[569,3110,598],{"className":3111},[630,631],[569,3113,605],{"className":3114,"style":652},[630,631],[569,3116],{"className":3117,"style":1265},[647],[569,3119,1060],{"className":3120},[1269],[569,3122],{"className":3123,"style":1265},[647],[569,3125,3127,3130],{"className":3126},[621],[569,3128],{"className":3129,"style":896},[625],[569,3131,3075],{"className":3132},[630],":",[3135,3136,3137,3270],"ul",{},[662,3138,3139,3140,1642,3191,3269],{},"In ",[569,3141,3143,3164],{"className":3142},[572],[569,3144,3146],{"className":3145},[576],[578,3147,3148],{"xmlns":580},[582,3149,3150,3162],{},[585,3151,3152,3154,3156,3158,3160],{},[592,3153,595],{"stretchy":594},[694,3155,696],{},[592,3157,602],{"separator":601},[694,3159,696],{},[592,3161,608],{"stretchy":594},[610,3163,2644],{"encoding":612},[569,3165,3167],{"className":3166,"ariaHidden":601},[617],[569,3168,3170,3173,3176,3179,3182,3185,3188],{"className":3169},[621],[569,3171],{"className":3172,"style":626},[625],[569,3174,595],{"className":3175},[636],[569,3177,696],{"className":3178},[630],[569,3180,602],{"className":3181},[643],[569,3183],{"className":3184,"style":648},[647],[569,3186,696],{"className":3187},[630],[569,3189,608],{"className":3190},[656],[569,3192,3194,3218],{"className":3193},[572],[569,3195,3197],{"className":3196},[576],[578,3198,3199],{"xmlns":580},[582,3200,3201,3215],{},[585,3202,3203,3205,3207,3209,3211,3213],{},[588,3204,1020],{},[592,3206,689],{},[592,3208,1060],{},[694,3210,3075],{},[592,3212,1492],{},[694,3214,696],{},[610,3216,3217],{"encoding":612},"D = -9 \u003C 0",[569,3219,3221,3239,3260],{"className":3220,"ariaHidden":601},[617],[569,3222,3224,3227,3230,3233,3236],{"className":3223},[621],[569,3225],{"className":3226,"style":1087},[625],[569,3228,1020],{"className":3229,"style":1091},[630,631],[569,3231],{"className":3232,"style":722},[647],[569,3234,689],{"className":3235},[726],[569,3237],{"className":3238,"style":722},[647],[569,3240,3242,3245,3248,3251,3254,3257],{"className":3241},[621],[569,3243],{"className":3244,"style":3041},[625],[569,3246,1060],{"className":3247},[630],[569,3249,3075],{"className":3250},[630],[569,3252],{"className":3253,"style":722},[647],[569,3255,1492],{"className":3256},[726],[569,3258],{"className":3259,"style":722},[647],[569,3261,3263,3266],{"className":3262},[621],[569,3264],{"className":3265,"style":896},[625],[569,3267,696],{"className":3268},[630],": Sattelpunkt.",[662,3271,3139,3272,1642,3323,900,3398,3519,3520,999],{},[569,3273,3275,3296],{"className":3274},[572],[569,3276,3278],{"className":3277},[576],[578,3279,3280],{"xmlns":580},[582,3281,3282,3294],{},[585,3283,3284,3286,3288,3290,3292],{},[592,3285,595],{"stretchy":594},[694,3287,2585],{},[592,3289,602],{"separator":601},[694,3291,2585],{},[592,3293,608],{"stretchy":594},[610,3295,2696],{"encoding":612},[569,3297,3299],{"className":3298,"ariaHidden":601},[617],[569,3300,3302,3305,3308,3311,3314,3317,3320],{"className":3301},[621],[569,3303],{"className":3304,"style":626},[625],[569,3306,595],{"className":3307},[636],[569,3309,2585],{"className":3310},[630],[569,3312,602],{"className":3313},[643],[569,3315],{"className":3316,"style":648},[647],[569,3318,2585],{"className":3319},[630],[569,3321,608],{"className":3322},[656],[569,3324,3326,3349],{"className":3325},[572],[569,3327,3329],{"className":3328},[576],[578,3330,3331],{"xmlns":580},[582,3332,3333,3346],{},[585,3334,3335,3337,3339,3342,3344],{},[588,3336,1020],{},[592,3338,689],{},[694,3340,3341],{},"27",[592,3343,1360],{},[694,3345,696],{},[610,3347,3348],{"encoding":612},"D = 27 > 0",[569,3350,3352,3370,3389],{"className":3351,"ariaHidden":601},[617],[569,3353,3355,3358,3361,3364,3367],{"className":3354},[621],[569,3356],{"className":3357,"style":1087},[625],[569,3359,1020],{"className":3360,"style":1091},[630,631],[569,3362],{"className":3363,"style":722},[647],[569,3365,689],{"className":3366},[726],[569,3368],{"className":3369,"style":722},[647],[569,3371,3373,3377,3380,3383,3386],{"className":3372},[621],[569,3374],{"className":3375,"style":3376},[625],"height:0.6835em;vertical-align:-0.0391em;",[569,3378,3341],{"className":3379},[630],[569,3381],{"className":3382,"style":722},[647],[569,3384,1360],{"className":3385},[726],[569,3387],{"className":3388,"style":722},[647],[569,3390,3392,3395],{"className":3391},[621],[569,3393],{"className":3394,"style":896},[625],[569,3396,696],{"className":3397},[630],[569,3399,3401,3431],{"className":3400},[572],[569,3402,3404],{"className":3403},[576],[578,3405,3406],{"xmlns":580},[582,3407,3408,3428],{},[585,3409,3410,3420,3422,3424,3426],{},[807,3411,3412,3414],{},[588,3413,590],{},[585,3415,3416,3418],{},[588,3417,598],{},[588,3419,598],{},[592,3421,689],{},[694,3423,2753],{},[592,3425,1360],{},[694,3427,696],{},[610,3429,3430],{"encoding":612},"f_{xx} = 6 > 0",[569,3432,3434,3492,3510],{"className":3433,"ariaHidden":601},[617],[569,3435,3437,3440,3483,3486,3489],{"className":3436},[621],[569,3438],{"className":3439,"style":712},[625],[569,3441,3443,3446],{"className":3442},[630],[569,3444,590],{"className":3445,"style":632},[630,631],[569,3447,3449],{"className":3448},[838],[569,3450,3452,3475],{"className":3451},[744,842],[569,3453,3455,3472],{"className":3454},[748],[569,3456,3458],{"className":3457,"style":849},[752],[569,3459,3460,3463],{"style":852},[569,3461],{"className":3462,"style":856},[759],[569,3464,3466],{"className":3465},[860,861,862,863],[569,3467,3469],{"className":3468},[630,863],[569,3470,1204],{"className":3471},[630,631,863],[569,3473,871],{"className":3474},[870],[569,3476,3478],{"className":3477},[748],[569,3479,3481],{"className":3480,"style":878},[752],[569,3482],{},[569,3484],{"className":3485,"style":722},[647],[569,3487,689],{"className":3488},[726],[569,3490],{"className":3491,"style":722},[647],[569,3493,3495,3498,3501,3504,3507],{"className":3494},[621],[569,3496],{"className":3497,"style":3376},[625],[569,3499,2753],{"className":3500},[630],[569,3502],{"className":3503,"style":722},[647],[569,3505,1360],{"className":3506},[726],[569,3508],{"className":3509,"style":722},[647],[569,3511,3513,3516],{"className":3512},[621],[569,3514],{"className":3515,"style":896},[625],[569,3517,696],{"className":3518},[630],": lokales Minimum mit ",[569,3521,3523,3553],{"className":3522},[572],[569,3524,3526],{"className":3525},[576],[578,3527,3528],{"xmlns":580},[582,3529,3530,3550],{},[585,3531,3532,3534,3536,3538,3540,3542,3544,3546,3548],{},[588,3533,590],{},[592,3535,595],{"stretchy":594},[694,3537,2585],{},[592,3539,602],{"separator":601},[694,3541,2585],{},[592,3543,608],{"stretchy":594},[592,3545,689],{},[592,3547,1060],{},[694,3549,2585],{},[610,3551,3552],{"encoding":612},"f(1, 1) = 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y) = x^4 + y^4",[569,3650,3652,3688,3732],{"className":3651,"ariaHidden":601},[617],[569,3653,3655,3658,3661,3664,3667,3670,3673,3676,3679,3682,3685],{"className":3654},[621],[569,3656],{"className":3657,"style":626},[625],[569,3659,590],{"className":3660,"style":632},[630,631],[569,3662,595],{"className":3663},[636],[569,3665,598],{"className":3666},[630,631],[569,3668,602],{"className":3669},[643],[569,3671],{"className":3672,"style":648},[647],[569,3674,605],{"className":3675,"style":652},[630,631],[569,3677,608],{"className":3678},[656],[569,3680],{"className":3681,"style":722},[647],[569,3683,689],{"className":3684},[726],[569,3686],{"className":3687,"style":722},[647],[569,3689,3691,3694,3723,3726,3729],{"className":3690},[621],[569,3692],{"className":3693,"style":1756},[625],[569,3695,3697,3700],{"className":3696},[630],[569,3698,598],{"className":3699},[630,631],[569,3701,3703],{"className":3702},[838],[569,3704,3706],{"className":3705},[744],[569,3707,3709],{"className":3708},[748],[569,3710,3712],{"className":3711,"style":1298},[752],[569,3713,3714,3717],{"style":1319},[569,3715],{"className":3716,"style":856},[759],[569,3718,3720],{"className":3719},[860,861,862,863],[569,3721,2450],{"className":3722},[630,863],[569,3724],{"className":3725,"style":1265},[647],[569,3727,1701],{"className":3728},[1269],[569,3730],{"className":3731,"style":1265},[647],[569,3733,3735,3738],{"className":3734},[621],[569,3736],{"className":3737,"style":1826},[625],[569,3739,3741,3744],{"className":3740},[630],[569,3742,605],{"className":3743,"style":652},[630,631],[569,3745,3747],{"className":3746},[838],[569,3748,3750],{"className":3749},[744],[569,3751,3753],{"className":3752},[748],[569,3754,3756],{"className":3755,"style":1298},[752],[569,3757,3758,3761],{"style":1319},[569,3759],{"className":3760,"style":856},[759],[569,3762,3764],{"className":3763},[860,861,862,863],[569,3765,2450],{"className":3766},[630,863],[569,3768,3770,3791],{"className":3769},[572],[569,3771,3773],{"className":3772},[576],[578,3774,3775],{"xmlns":580},[582,3776,3777,3789],{},[585,3778,3779,3781,3783,3785,3787],{},[592,3780,595],{"stretchy":594},[694,3782,696],{},[592,3784,602],{"separator":601},[694,3786,696],{},[592,3788,608],{"stretchy":594},[610,3790,2644],{"encoding":612},[569,3792,3794],{"className":3793,"ariaHidden":601},[617],[569,3795,3797,3800,3803,3806,3809,3812,3815],{"className":3796},[621],[569,3798],{"className":3799,"style":626},[625],[569,3801,595],{"className":3802},[636],[569,3804,696],{"className":3805},[630],[569,3807,602],{"className":3808},[643],[569,3810],{"className":3811,"style":648},[647],[569,3813,696],{"className":3814},[630],[569,3816,608],{"className":3817},[656]," kritischer Punkt mit ",[569,3820,3822,3839],{"className":3821},[572],[569,3823,3825],{"className":3824},[576],[578,3826,3827],{"xmlns":580},[582,3828,3829,3837],{},[585,3830,3831,3833,3835],{},[588,3832,1020],{},[592,3834,689],{},[694,3836,696],{},[610,3838,1611],{"encoding":612},[569,3840,3842,3860],{"className":3841,"ariaHidden":601},[617],[569,3843,3845,3848,3851,3854,3857],{"className":3844},[621],[569,3846],{"className":3847,"style":1087},[625],[569,3849,1020],{"className":3850,"style":1091},[630,631],[569,3852],{"className":3853,"style":722},[647],[569,3855,689],{"className":3856},[726],[569,3858],{"className":3859,"style":722},[647],[569,3861,3863,3866],{"className":3862},[621],[569,3864],{"className":3865,"style":896},[625],[569,3867,696],{"className":3868},[630],". Der Test versagt, obwohl dort offensichtlich ein Minimum liegt, denn ",[569,3871,3873,3906],{"className":3872},[572],[569,3874,3876],{"className":3875},[576],[578,3877,3878],{"xmlns":580},[582,3879,3880,3903],{},[585,3881,3882,3884,3887,3889,3891,3893,3895,3897,3899,3901],{},[588,3883,590],{},[592,3885,3886],{},"≥",[694,3888,696],{},[592,3890,689],{},[588,3892,590],{},[592,3894,595],{"stretchy":594},[694,3896,696],{},[592,3898,602],{"separator":601},[694,3900,696],{},[592,3902,608],{"stretchy":594},[610,3904,3905],{"encoding":612},"f \\geq 0 = f(0, 0)",[569,3907,3909,3927,3945],{"className":3908,"ariaHidden":601},[617],[569,3910,3912,3915,3918,3921,3924],{"className":3911},[621],[569,3913],{"className":3914,"style":712},[625],[569,3916,590],{"className":3917,"style":632},[630,631],[569,3919],{"className":3920,"style":722},[647],[569,3922,3886],{"className":3923},[726],[569,3925],{"className":3926,"style":722},[647],[569,3928,3930,3933,3936,3939,3942],{"className":3929},[621],[569,3931],{"className":3932,"style":896},[625],[569,3934,696],{"className":3935},[630],[569,3937],{"className":3938,"style":722},[647],[569,3940,689],{"className":3941},[726],[569,3943],{"className":3944,"style":722},[647],[569,3946,3948,3951,3954,3957,3960,3963,3966,3969],{"className":3947},[621],[569,3949],{"className":3950,"style":626},[625],[569,3952,590],{"className":3953,"style":632},[630,631],[569,3955,595],{"className":3956},[636],[569,3958,696],{"className":3959},[630],[569,3961,602],{"className":3962},[643],[569,3964],{"className":3965,"style":648},[647],[569,3967,696],{"className":3968},[630],[569,3970,608],{"className":3971},[656],[560,3973,3975],{"id":3974},"typische-fehler","Typische Fehler",[3135,3977,3978,4108,4162,4337],{},[662,3979,1343,3980,4030,4031,4107],{},[569,3981,3983,4000],{"className":3982},[572],[569,3984,3986],{"className":3985},[576],[578,3987,3988],{"xmlns":580},[582,3989,3990,3998],{},[585,3991,3992,3994,3996],{},[588,3993,1020],{},[592,3995,1360],{},[694,3997,696],{},[610,3999,1365],{"encoding":612},[569,4001,4003,4021],{"className":4002,"ariaHidden":601},[617],[569,4004,4006,4009,4012,4015,4018],{"className":4005},[621],[569,4007],{"className":4008,"style":1375},[625],[569,4010,1020],{"className":4011,"style":1091},[630,631],[569,4013],{"className":4014,"style":722},[647],[569,4016,1360],{"className":4017},[726],[569,4019],{"className":4020,"style":722},[647],[569,4022,4024,4027],{"className":4023},[621],[569,4025],{"className":4026,"style":896},[625],[569,4028,696],{"className":4029},[630]," das Vorzeichen von ",[569,4032,4034,4055],{"className":4033},[572],[569,4035,4037],{"className":4036},[576],[578,4038,4039],{"xmlns":580},[582,4040,4041,4053],{},[585,4042,4043],{},[807,4044,4045,4047],{},[588,4046,590],{},[585,4048,4049,4051],{},[588,4050,598],{},[588,4052,598],{},[610,4054,1422],{"encoding":612},[569,4056,4058],{"className":4057,"ariaHidden":601},[617],[569,4059,4061,4064],{"className":4060},[621],[569,4062],{"className":4063,"style":712},[625],[569,4065,4067,4070],{"className":4066},[630],[569,4068,590],{"className":4069,"style":632},[630,631],[569,4071,4073],{"className":4072},[838],[569,4074,4076,4099],{"className":4075},[744,842],[569,4077,4079,4096],{"className":4078},[748],[569,4080,4082],{"className":4081,"style":849},[752],[569,4083,4084,4087],{"style":852},[569,4085],{"className":4086,"style":856},[759],[569,4088,4090],{"className":4089},[860,861,862,863],[569,4091,4093],{"className":4092},[630,863],[569,4094,1204],{"className":4095},[630,631,863],[569,4097,871],{"className":4098},[870],[569,4100,4102],{"className":4101},[748],[569,4103,4105],{"className":4104,"style":878},[752],[569,4106],{}," nicht prüfen und Minimum und Maximum verwechseln.",[662,4109,4110,4111,4161],{},"Aus ",[569,4112,4114,4131],{"className":4113},[572],[569,4115,4117],{"className":4116},[576],[578,4118,4119],{"xmlns":580},[582,4120,4121,4129],{},[585,4122,4123,4125,4127],{},[588,4124,1020],{},[592,4126,689],{},[694,4128,696],{},[610,4130,1611],{"encoding":612},[569,4132,4134,4152],{"className":4133,"ariaHidden":601},[617],[569,4135,4137,4140,4143,4146,4149],{"className":4136},[621],[569,4138],{"className":4139,"style":1087},[625],[569,4141,1020],{"className":4142,"style":1091},[630,631],[569,4144],{"className":4145,"style":722},[647],[569,4147,689],{"className":4148},[726],[569,4150],{"className":4151,"style":722},[647],[569,4153,4155,4158],{"className":4154},[621],[569,4156],{"className":4157,"style":896},[625],[569,4159,696],{"className":4160},[630]," schließen, dass kein Extremum vorliegt. Der Test macht dann gar keine Aussage.",[662,4163,4164,4165,4256,4257,4285,4286,4336],{},"Beim Lösen von ",[569,4166,4168,4191],{"className":4167},[572],[569,4169,4171],{"className":4170},[576],[578,4172,4173],{"xmlns":580},[582,4174,4175,4189],{},[585,4176,4177,4179,4181,4183],{},[588,4178,684],{"mathvariant":683},[588,4180,590],{},[592,4182,689],{},[691,4184,4185,4187],{"accent":601},[694,4186,696],{},[592,4188,699],{},[610,4190,702],{"encoding":612},[569,4192,4194,4215],{"className":4193,"ariaHidden":601},[617],[569,4195,4197,4200,4203,4206,4209,4212],{"className":4196},[621],[569,4198],{"className":4199,"style":712},[625],[569,4201,684],{"className":4202},[630],[569,4204,590],{"className":4205,"style":632},[630,631],[569,4207],{"className":4208,"style":722},[647],[569,4210,689],{"className":4211},[726],[569,4213],{"className":4214,"style":722},[647],[569,4216,4218,4221],{"className":4217},[621],[569,4219],{"className":4220,"style":736},[625],[569,4222,4224],{"className":4223},[630,740],[569,4225,4227],{"className":4226},[744],[569,4228,4230],{"className":4229},[748],[569,4231,4233,4241],{"className":4232,"style":736},[752],[569,4234,4235,4238],{"style":755},[569,4236],{"className":4237,"style":760},[759],[569,4239,696],{"className":4240},[630],[569,4242,4243,4246],{"style":766},[569,4244],{"className":4245,"style":760},[759],[569,4247,4249],{"className":4248,"style":774},[773],[569,4250,4252],{"className":4251,"style":779},[778],[781,4253,4254],{"xmlns":783,"width":784,"height":785,"style":786,"viewBox":787,"preserveAspectRatio":788},[790,4255],{"d":792}," Lösungen verlieren, z.B. durch Teilen durch ",[569,4258,4260,4273],{"className":4259},[572],[569,4261,4263],{"className":4262},[576],[578,4264,4265],{"xmlns":580},[582,4266,4267,4271],{},[585,4268,4269],{},[588,4270,598],{},[610,4272,598],{"encoding":612},[569,4274,4276],{"className":4275,"ariaHidden":601},[617],[569,4277,4279,4282],{"className":4278},[621],[569,4280],{"className":4281,"style":2511},[625],[569,4283,598],{"className":4284},[630,631],", ohne den Fall ",[569,4287,4289,4306],{"className":4288},[572],[569,4290,4292],{"className":4291},[576],[578,4293,4294],{"xmlns":580},[582,4295,4296,4304],{},[585,4297,4298,4300,4302],{},[588,4299,598],{},[592,4301,689],{},[694,4303,696],{},[610,4305,2535],{"encoding":612},[569,4307,4309,4327],{"className":4308,"ariaHidden":601},[617],[569,4310,4312,4315,4318,4321,4324],{"className":4311},[621],[569,4313],{"className":4314,"style":2511},[625],[569,4316,598],{"className":4317},[630,631],[569,4319],{"className":4320,"style":722},[647],[569,4322,689],{"className":4323},[726],[569,4325],{"className":4326,"style":722},[647],[569,4328,4330,4333],{"className":4329},[621],[569,4331],{"className":4332,"style":896},[625],[569,4334,696],{"className":4335},[630]," zu betrachten.",[662,4338,4339,4432,4433,4461],{},[569,4340,4342,4366],{"className":4341},[572],[569,4343,4345],{"className":4344},[576],[578,4346,4347],{"xmlns":580},[582,4348,4349,4363],{},[585,4350,4351],{},[1062,4352,4353,4355,4361],{},[588,4354,590],{},[585,4356,4357,4359],{},[588,4358,598],{},[588,4360,605],{},[694,4362,1074],{},[610,4364,4365],{"encoding":612},"f_{xy}^2",[569,4367,4369],{"className":4368,"ariaHidden":601},[617],[569,4370,4372,4375],{"className":4371},[621],[569,4373],{"className":4374,"style":1279},[625],[569,4376,4378,4381],{"className":4377},[630],[569,4379,590],{"className":4380,"style":632},[630,631],[569,4382,4384],{"className":4383},[838],[569,4385,4387,4424],{"className":4386},[744,842],[569,4388,4390,4421],{"className":4389},[748],[569,4391,4393,4410],{"className":4392,"style":1298},[752],[569,4394,4395,4398],{"style":1301},[569,4396],{"className":4397,"style":856},[759],[569,4399,4401],{"className":4400},[860,861,862,863],[569,4402,4404,4407],{"className":4403},[630,863],[569,4405,598],{"className":4406},[630,631,863],[569,4408,605],{"className":4409,"style":652},[630,631,863],[569,4411,4412,4415],{"style":1319},[569,4413],{"className":4414,"style":856},[759],[569,4416,4418],{"className":4417},[860,861,862,863],[569,4419,1074],{"className":4420},[630,863],[569,4422,871],{"className":4423},[870],[569,4425,4427],{"className":4426},[748],[569,4428,4430],{"className":4429,"style":1338},[752],[569,4431],{}," in ",[569,4434,4436,4449],{"className":4435},[572],[569,4437,4439],{"className":4438},[576],[578,4440,4441],{"xmlns":580},[582,4442,4443,4447],{},[585,4444,4445],{},[588,4446,1020],{},[610,4448,1020],{"encoding":612},[569,4450,4452],{"className":4451,"ariaHidden":601},[617],[569,4453,4455,4458],{"className":4454},[621],[569,4456],{"className":4457,"style":1087},[625],[569,4459,1020],{"className":4460,"style":1091},[630,631]," vergessen oder mit falschem Vorzeichen einsetzen.",{"title":4463,"searchDepth":4464,"depth":4464,"links":4465},"",2,[4466,4467,4468],{"id":562,"depth":4464,"text":563},{"id":1652,"depth":4464,"text":1653},{"id":3974,"depth":4464,"text":3975},[4470,4472,4476,4479,4482,4486,4488,4492],{"lhs":4471,"rhs":696},"round(((x + 1e-5)^3 - 3*(x + 1e-5)*y + y^3 - ((x - 1e-5)^3 - 3*(x - 1e-5)*y + y^3)) \u002F (2e-5) - (3*x^2 - 3*y), 6)",{"lhs":4473,"rhs":696,"where":4474},"3*x^2 - 3*y",{"x":4475,"y":4475},0,{"lhs":4473,"rhs":696,"where":4477},{"x":4478,"y":4478},1,{"lhs":4480,"rhs":696,"where":4481},"3*y^2 - 3*x",{"x":4478,"y":4478},{"lhs":4483,"rhs":4484,"where":4485},"6*x*6*y - (-3)^2","-9",{"x":4475,"y":4475},{"lhs":4483,"rhs":3341,"where":4487},{"x":4478,"y":4478},{"lhs":4489,"rhs":4490,"where":4491},"x^3 - 3*x*y + y^3","-1",{"x":4478,"y":4478},{"lhs":4493,"rhs":696,"where":4494},"12*x^2 * 12*y^2 - 0^2",{"x":4475,"y":4475},"f \\text{ zweimal stetig differenzierbar in einer Umgebung von } (x_0, y_0),\\ \\nabla f(x_0, y_0) = \\vec{0},\\ \\text{Ableitungen an der Stelle } (x_0, y_0)","\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 stetig differenzierbar in einer Umgebung von\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(x_0, y_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:1em;vertical-align:-0.25em;\">\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=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\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:-0.0359em;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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class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2134em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2355em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Ableitungen an der Stelle\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(x_0, y_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:1em;vertical-align:-0.25em;\">\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 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class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","multivariable-extrema","D = f_{xx} f_{yy} - f_{xy}^2 \\quad \\begin{cases} D > 0,\\ f_{xx} > 0: & \\text{lokales Minimum} \\\\ D > 0,\\ f_{xx} \u003C 0: & \\text{lokales Maximum} \\\\ D \u003C 0: & \\text{Sattelpunkt} \\end{cases}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>y\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsubsup>\u003Cmspace width=\"1em\"\u002F>\u003Cmrow>\u003Cmo fence=\"true\">{\u003C\u002Fmo>\u003Cmtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\">\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>:\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmtext>lokales Minimum\u003C\u002Fmtext>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo><\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>:\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmtext>lokales Maximum\u003C\u002Fmtext>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003Cmtr>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003Cmo><\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>:\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003Cmtd>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmtext>Sattelpunkt\u003C\u002Fmtext>\u003C\u002Fmstyle>\u003C\u002Fmtd>\u003C\u002Fmtr>\u003C\u002Fmtable>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">D = f_{xx} f_{yy} - f_{xy}^2 \\quad \\begin{cases} D > 0,\\ f_{xx} > 0: & \\text{lokales Minimum} \\\\ D > 0,\\ f_{xx} < 0: & \\text{lokales Maximum} \\\\ D < 0: & \\text{Sattelpunkt} \\end{cases}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan 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mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">xx\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:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1076em;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 mathnormal mtight\" 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style=\"height:0.15em;\">\u003Cspan>\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=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">:\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-1.53em;\">\u003Cspan class=\"pstrut\" style=\"height:3.008em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\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=\"mrel\">:\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:1.91em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"arraycolsep\" style=\"width:1em;\">\u003C\u002Fspan>\u003Cspan class=\"col-align-l\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:2.41em;\">\u003Cspan style=\"top:-4.41em;\">\u003Cspan class=\"pstrut\" style=\"height:3.008em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">lokales Minimum\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.97em;\">\u003Cspan class=\"pstrut\" style=\"height:3.008em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">lokales Maximum\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-1.53em;\">\u003Cspan class=\"pstrut\" style=\"height:3.008em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">Sattelpunkt\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:1.91em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">D\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:0.0278em;\">D\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4526,"meaning":4527,"symbolHtml":4528},"f_{xx}, f_{yy}","zweite partielle Ableitungen nach x bzw. nach y","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>f\u003C\u002Fmi>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo 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