大阪大学 前期理系 1980年度 問5

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 1980年度
問No 問5
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} \usepackage{vector3} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  次の条件を満たす関数 $f(x)\,\,\,(x > 0)$ を求めよ.  曲線 $C : y = f(x)\,\,\,(x > 0)$ 上の任意の点Pにおける接線と$x$軸との交点をQとすると,線分PQは$y$軸によって2等分される. また, 点$(0,\,\,3)$と $C$ 上の点との距離の最小値は$\sqrt{\vphantom{b} 2}$である. \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \fbox{解答}} \vskip 2mm $\P(t,\,\,f(t))\,\,\,(t > 0)$とおく. Pにおける $C$ の接線は, \begin{align*} y = f'(t)(x - t) + f(t) \tag*{$\cdott\MARU{1}$} \end{align*} $y$軸が線分PQを二等分することから, $\Q(-t,\,\,0)$とおける. Qは\MARU{1}上の点だから, \begin{align*} 0 = f'(t)(-t - t) + f(t) = 0 \qquad \therefore \,\,\, f'(t) - \frac{1}{2t}f(t) = 0 \tag*{$\cdott\MARU{2}$} \end{align*} $\displaystyle\int \left({-\dfrac{1}{2t}}\right)dt = -\dfrac{1}{2}\log t + A\,\,\,(Aは定数)$ に留意して,\smallskip \MARU{2}の両辺に $e^{-\frac{1}{2}\log t} = \dfrac{1}{\sqrt{\vphantom{b} t}}$ % を掛ければ,\\ \begin{minipage}{240pt} \begin{gather*} \frac{1}{\sqrt{\vphantom{b} t}}f'(t) - \frac{2}{t\sqrt{\vphantom{b} t}}f(t) = 0 \\[1mm] \left\{\frac{1}{\sqrt{\vphantom{b} t}}f(t) \right\}^{\! \prime} = 0 \quad \left(\because\,\,\, \left(\frac{1}{\sqrt{\vphantom{b} t}} \right)^{\!\! \prime} = -\frac{2}{t\sqrt{\vphantom{b} t}} \right)\\[1mm] \frac{1}{\sqrt{\vphantom{b} t}}f(t) = B \quad (Bは定数) \\ \therefore \,\,\, f(x) = B\sqrt{\vphantom{b} x} \quad\mbox{\footnotesize$(tをxにとり換えた)$} \end{gather*} \end{minipage} \begin{minipage}{140pt} %\input{osaka80s5_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 22.6800, 14.2000)( 4.0000,-20.2000) % STR 2 0 3 0 % 3 1500 586 1500 600 4 2400 % $y$ \put(15.0000,-6.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2670 1727 2670 1740 4 2400 % $x$ \put(26.7000,-17.4000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1534 2020 1534 600 % \special{pn 8}% \special{pa 1534 2020}% \special{pa 1534 600}% \special{fp}% \special{sh 1}% \special{pa 1534 600}% \special{pa 1514 668}% \special{pa 1534 654}% \special{pa 1554 668}% \special{pa 1534 600}% \special{fp}% % VECTOR 2 0 3 0 % 2 400 1696 2668 1696 % \special{pn 8}% \special{pa 400 1696}% \special{pa 2668 1696}% \special{fp}% \special{sh 1}% \special{pa 2668 1696}% \special{pa 2602 1676}% \special{pa 2616 1696}% \special{pa 2602 1716}% \special{pa 2668 1696}% \special{fp}% % DOT 0 0 3 0 % 1 2182 1048 % \special{pn 20}% \special{sh 1}% \special{ar 2182 1048 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 886 1696 % \special{pn 20}% \special{sh 1}% \special{ar 886 1696 10 10 0 6.28318530717959E+0000}% % STR 2 0 3 0 % 3 2210 1099 2210 1180 2 0 % {\scriptsize$\P(t,\,f(t))$} \put(22.1000,-11.8000){\makebox(0,0)[lb]{{\scriptsize$\P(t,\,f(t))$}}}% % STR 2 0 3 0 % 3 460 1569 460 1650 2 0 % {\scriptsize$\Q(-t,\,0)$} \put(4.6000,-16.5000){\makebox(0,0)[lb]{{\scriptsize$\Q(-t,\,0)$}}}% % LINE 2 2 3 0 % 2 2182 1696 2182 1048 % \special{pn 8}% \special{pa 2182 1696}% \special{pa 2182 1048}% \special{dt 0.045}% % LINE 2 0 3 0 % 2 400 1939 2668 810 % \special{pn 8}% \special{pa 400 1940}% \special{pa 2668 810}% \special{fp}% % CIRCLE 3 0 3 0 % 4 1210 1061 1218 1773 826 1809 1591 1806 % \special{pn 4}% \special{ar 1210 1062 712 712 1.0980629 2.0450818}% % CIRCLE 3 0 3 0 % 4 1858 1061 1866 1773 1474 1809 2239 1806 % \special{pn 4}% \special{ar 1858 1062 712 712 1.0980629 2.0450818}% % LINE 0 0 3 0 % 2 1194 1741 1194 1798 % \special{pn 20}% \special{pa 1194 1742}% \special{pa 1194 1798}% \special{fp}% % LINE 0 0 3 0 % 2 1218 1741 1218 1798 % \special{pn 20}% \special{pa 1218 1742}% \special{pa 1218 1798}% \special{fp}% % LINE 0 0 3 0 % 2 1846 1741 1846 1798 % \special{pn 20}% \special{pa 1846 1742}% \special{pa 1846 1798}% \special{fp}% % LINE 0 0 3 0 % 2 1871 1741 1871 1798 % \special{pn 20}% \special{pa 1872 1742}% \special{pa 1872 1798}% \special{fp}% % CIRCLE 3 0 3 0 % 4 1583 2279 2486 2219 1502 862 615 1461 % \special{pn 4}% \special{ar 1584 2280 906 906 3.8432009 4.6552881}% % CIRCLE 3 0 3 0 % 4 2231 1955 3134 1895 2150 538 1263 1137 % \special{pn 4}% \special{ar 2232 1956 906 906 3.8432009 4.6552881}% % CIRCLE 1 0 3 0 % 4 1186 1466 1186 1494 1186 1494 1186 1494 % \special{pn 13}% \special{ar 1186 1466 28 28 0.0000000 6.2831853}% % CIRCLE 1 0 3 0 % 4 1838 1142 1838 1170 1838 1170 1838 1170 % \special{pn 13}% \special{ar 1838 1142 28 28 0.0000000 6.2831853}% % SPLINE 1 0 3 0 % 14 1535 1630 1590 1500 1680 1400 1785 1290 1895 1220 2015 1140 2100 1095 2185 1050 2280 1005 2385 965 2450 935 2575 895 2635 875 2660 875 % \special{pn 13}% \special{pa 1536 1630}% \special{pa 1546 1600}% \special{pa 1556 1568}% \special{pa 1568 1540}% \special{pa 1584 1512}% \special{pa 1602 1486}% \special{pa 1622 1462}% \special{pa 1644 1438}% \special{pa 1666 1414}% \special{pa 1688 1392}% \special{pa 1710 1368}% \special{pa 1732 1344}% \special{pa 1754 1320}% \special{pa 1776 1298}% \special{pa 1802 1278}% \special{pa 1828 1260}% \special{pa 1856 1244}% \special{pa 1884 1228}% \special{pa 1910 1210}% \special{pa 1938 1192}% \special{pa 1964 1174}% \special{pa 1990 1156}% \special{pa 2016 1140}% \special{pa 2044 1124}% \special{pa 2074 1110}% \special{pa 2102 1094}% \special{pa 2130 1080}% \special{pa 2158 1064}% \special{pa 2186 1050}% \special{pa 2216 1036}% \special{pa 2244 1022}% \special{pa 2274 1008}% \special{pa 2304 996}% \special{pa 2334 986}% \special{pa 2364 974}% \special{pa 2392 962}% \special{pa 2422 948}% \special{pa 2450 936}% \special{pa 2480 924}% \special{pa 2512 916}% \special{pa 2542 908}% \special{pa 2572 896}% \special{pa 2602 884}% \special{pa 2634 876}% \special{pa 2660 876}% \special{sp}% % STR 2 0 3 0 % 3 1410 1750 1410 1850 2 0 % {\footnotesize O} \put(14.1000,-18.5000){\makebox(0,0)[lb]{{\footnotesize O}}}% % STR 2 0 3 0 % 3 1660 1420 1660 1520 2 0 % {\scriptsize $C$} \put(16.6000,-15.2000){\makebox(0,0)[lb]{{\scriptsize $C$}}}% \end{picture}% \end{minipage} \vskip 0.5zw \noindent% $B \leqq 0$ ならば点$(0,\,\,3)$と $C$ 上の点との距離の最小値は $3\,\,(> \sqrt{\vphantom{b} 2})$となり矛盾. ゆえに $B > 0$ である. $(0,\,\,3)$との距離の最小値を与える $C$ 上の点を$\R(u^2,\,\,Bu)\,\,\, (u > 0)$とおく. Rは中心$\I(0,\,\,3)$,半径$\sqrt{\vphantom{b} 2}$の円 $K$ と $C$ の接点である. Rが $K$ 上にあることから,\\ \begin{minipage}{230pt} \begin{align*} u^4 + (Bu - 3)^2 = 2 \tag*{$\cdott\MARU{3}$} \end{align*}  \smallskip$\bekutoru{$\I\R$}$ はRにおける $C$ の接線に垂直であり, この接線はベクトル \smallskip$(1,\,\,f'(u^2)) = \dfrac{1}{2u}(2u,\,\,B)$ に 平行だから, \begin{gather*} (u^2,\,\,Bu - 3) \cdot (2u,\,\,B) = 0 \\ \therefore \,\,\, 2u^3 + B(Bu - 3) = 0 \tag*{$\cdots\cdotssp\MARU{4}$} \end{gather*} $\MARU{3} \times 2 - \MARU{4} \times u$より $Bu = v$ とおけば, \end{minipage} \begin{minipage}{140pt} \hspace*{1zw}% %\input{osaka80s5_zu_4} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 20.7400, 18.4000)( 4.0000,-22.4000) % STR 2 0 3 0 % 3 800 2199 800 2210 2 0 % {\footnotesize O} \put(8.0000,-22.1000){\makebox(0,0)[lb]{{\footnotesize O}}}% % STR 2 0 3 0 % 3 892 519 892 530 4 3200 % $y$ \put(8.9200,-5.3000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2470 2109 2470 2120 4 3200 % $x$ \put(24.7000,-21.2000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 920 2240 920 526 % \special{pn 8}% \special{pa 920 2240}% \special{pa 920 526}% \special{fp}% \special{sh 1}% \special{pa 920 526}% \special{pa 900 594}% \special{pa 920 580}% \special{pa 940 594}% \special{pa 920 526}% \special{fp}% % VECTOR 2 0 3 0 % 2 400 2085 2474 2085 % \special{pn 8}% \special{pa 400 2086}% \special{pa 2474 2086}% \special{fp}% \special{sh 1}% \special{pa 2474 2086}% \special{pa 2408 2066}% \special{pa 2422 2086}% \special{pa 2408 2106}% \special{pa 2474 2086}% \special{fp}% % FUNC 2 0 3 0 % 9 789 400 2474 2214 918 2085 1178 2085 918 1826 918 400 2474 2214 0 5 0 0 % 2sqrt(x) \special{pn 8}% \special{pa 920 2040}% \special{pa 926 2000}% \special{pa 930 1974}% \special{pa 936 1954}% \special{pa 940 1934}% \special{pa 946 1918}% \special{pa 950 1904}% \special{pa 956 1890}% \special{pa 960 1878}% \special{pa 966 1866}% \special{pa 970 1854}% \special{pa 976 1842}% \special{pa 980 1832}% \special{pa 986 1822}% \special{pa 990 1812}% \special{pa 996 1804}% \special{pa 1000 1794}% \special{pa 1006 1786}% \special{pa 1010 1778}% \special{pa 1016 1770}% \special{pa 1020 1762}% \special{pa 1026 1754}% \special{pa 1030 1746}% \special{pa 1036 1738}% \special{pa 1040 1730}% \special{pa 1046 1724}% \special{pa 1050 1716}% \special{pa 1056 1710}% \special{pa 1060 1702}% \special{pa 1066 1696}% \special{pa 1070 1690}% \special{pa 1076 1682}% \special{pa 1080 1676}% \special{pa 1086 1670}% \special{pa 1090 1664}% \special{pa 1096 1658}% \special{pa 1100 1652}% \special{pa 1106 1646}% \special{pa 1110 1640}% \special{pa 1116 1634}% \special{pa 1120 1628}% \special{pa 1126 1624}% \special{pa 1130 1618}% \special{pa 1136 1612}% \special{pa 1140 1606}% \special{pa 1146 1602}% \special{pa 1150 1596}% \special{pa 1156 1590}% \special{pa 1160 1586}% \special{pa 1166 1580}% \special{pa 1170 1576}% \special{pa 1176 1570}% \special{pa 1180 1566}% \special{pa 1186 1560}% \special{pa 1190 1556}% \special{pa 1196 1550}% \special{pa 1200 1546}% \special{pa 1206 1542}% \special{pa 1210 1536}% \special{pa 1216 1532}% \special{pa 1220 1528}% \special{pa 1226 1522}% \special{pa 1230 1518}% \special{pa 1236 1514}% \special{pa 1240 1510}% \special{pa 1246 1504}% \special{pa 1250 1500}% \special{pa 1256 1496}% \special{pa 1260 1492}% \special{pa 1266 1488}% \special{pa 1270 1482}% \special{pa 1276 1478}% \special{pa 1280 1474}% \special{pa 1286 1470}% \special{pa 1290 1466}% \special{pa 1296 1462}% \special{pa 1300 1458}% \special{pa 1306 1454}% \special{pa 1310 1450}% \special{pa 1316 1446}% \special{pa 1320 1442}% \special{pa 1326 1438}% \special{pa 1330 1434}% \special{pa 1336 1430}% \special{pa 1340 1426}% \special{pa 1346 1422}% \special{pa 1350 1418}% \special{pa 1356 1414}% \special{pa 1360 1410}% \special{pa 1366 1406}% \special{pa 1370 1402}% \special{pa 1376 1398}% \special{pa 1380 1394}% \special{pa 1386 1392}% \special{pa 1390 1388}% \special{pa 1396 1384}% \special{pa 1400 1380}% \special{pa 1406 1376}% \special{pa 1410 1372}% \special{pa 1416 1370}% \special{pa 1420 1366}% \special{pa 1426 1362}% \special{pa 1430 1358}% \special{pa 1436 1356}% \special{pa 1440 1352}% \special{pa 1446 1348}% \special{pa 1450 1344}% \special{pa 1456 1342}% \special{pa 1460 1338}% \special{pa 1466 1334}% \special{pa 1470 1330}% \special{pa 1476 1328}% \special{pa 1480 1324}% \special{pa 1486 1320}% \special{pa 1490 1318}% \special{pa 1496 1314}% \special{pa 1500 1310}% \special{pa 1506 1308}% \special{pa 1510 1304}% \special{pa 1516 1300}% \special{pa 1520 1298}% \special{pa 1526 1294}% \special{pa 1530 1290}% \special{pa 1536 1288}% \special{pa 1540 1284}% \special{pa 1546 1282}% \special{pa 1550 1278}% \special{pa 1556 1274}% \special{pa 1560 1272}% \special{pa 1566 1268}% \special{pa 1570 1266}% \special{pa 1576 1262}% \special{pa 1580 1258}% \special{pa 1586 1256}% \special{pa 1590 1252}% \special{pa 1596 1250}% \special{pa 1600 1246}% \special{pa 1606 1244}% \special{pa 1610 1240}% \special{pa 1616 1238}% \special{pa 1620 1234}% \special{pa 1626 1232}% \special{pa 1630 1228}% \special{pa 1636 1226}% \special{pa 1640 1222}% \special{pa 1646 1220}% \special{pa 1650 1216}% \special{pa 1656 1214}% \special{pa 1660 1210}% \special{pa 1666 1208}% \special{pa 1670 1204}% \special{pa 1676 1202}% \special{pa 1680 1198}% \special{pa 1686 1196}% \special{pa 1690 1192}% \special{pa 1696 1190}% \special{pa 1700 1188}% \special{pa 1706 1184}% \special{pa 1710 1182}% \special{pa 1716 1178}% \special{pa 1720 1176}% \special{pa 1726 1172}% \special{pa 1730 1170}% \special{pa 1736 1168}% \special{pa 1740 1164}% \special{pa 1746 1162}% \special{pa 1750 1158}% \special{pa 1756 1156}% \special{pa 1760 1154}% \special{pa 1766 1150}% \special{pa 1770 1148}% \special{pa 1776 1146}% \special{pa 1780 1142}% \special{pa 1786 1140}% \special{pa 1790 1136}% \special{pa 1796 1134}% \special{pa 1800 1132}% \special{pa 1806 1128}% \special{pa 1810 1126}% \special{pa 1816 1124}% \special{pa 1820 1120}% \special{pa 1826 1118}% \special{pa 1830 1116}% \special{pa 1836 1112}% \special{pa 1840 1110}% \special{pa 1846 1108}% \special{pa 1850 1104}% \special{pa 1856 1102}% \special{pa 1860 1100}% \special{pa 1866 1096}% \special{pa 1870 1094}% \special{pa 1876 1092}% \special{pa 1880 1090}% \special{pa 1886 1086}% \special{pa 1890 1084}% \special{pa 1896 1082}% \special{pa 1900 1078}% \special{pa 1906 1076}% \special{pa 1910 1074}% \special{pa 1916 1072}% \special{pa 1920 1068}% \special{pa 1926 1066}% \special{pa 1930 1064}% \special{pa 1936 1062}% \special{pa 1940 1058}% \special{pa 1946 1056}% \special{pa 1950 1054}% \special{pa 1956 1050}% \special{pa 1960 1048}% \special{pa 1966 1046}% \special{pa 1970 1044}% \special{pa 1976 1042}% \special{pa 1980 1038}% \special{pa 1986 1036}% \special{pa 1990 1034}% \special{pa 1996 1032}% \special{pa 2000 1028}% \special{pa 2006 1026}% \special{pa 2010 1024}% \special{pa 2016 1022}% \special{pa 2020 1020}% \special{pa 2026 1016}% \special{pa 2030 1014}% \special{pa 2036 1012}% \special{pa 2040 1010}% \special{pa 2046 1008}% \special{pa 2050 1004}% \special{pa 2056 1002}% \special{pa 2060 1000}% \special{pa 2066 998}% \special{pa 2070 996}% \special{pa 2076 992}% \special{pa 2080 990}% \special{pa 2086 988}% \special{pa 2090 986}% \special{pa 2096 984}% \special{pa 2100 982}% \special{pa 2106 978}% \special{pa 2110 976}% \special{pa 2116 974}% \special{pa 2120 972}% \special{pa 2126 970}% \special{pa 2130 968}% \special{pa 2136 964}% \special{pa 2140 962}% \special{pa 2146 960}% \special{pa 2150 958}% \special{pa 2156 956}% \special{pa 2160 954}% \special{pa 2166 952}% \special{pa 2170 948}% \special{pa 2176 946}% \special{pa 2180 944}% \special{pa 2186 942}% \special{pa 2190 940}% \special{pa 2196 938}% \special{pa 2200 936}% \special{pa 2206 934}% \special{pa 2210 930}% \special{pa 2216 928}% \special{pa 2220 926}% \special{pa 2226 924}% \special{pa 2230 922}% \special{pa 2236 920}% \special{pa 2240 918}% \special{pa 2246 916}% \special{pa 2250 914}% \special{pa 2256 910}% \special{pa 2260 908}% \special{pa 2266 906}% \special{pa 2270 904}% \special{pa 2276 902}% \special{pa 2280 900}% \special{pa 2286 898}% \special{pa 2290 896}% \special{pa 2296 894}% \special{pa 2300 892}% \special{pa 2306 890}% \special{pa 2310 886}% \special{pa 2316 884}% \special{pa 2320 882}% \special{pa 2326 880}% \special{pa 2330 878}% \special{pa 2336 876}% \special{pa 2340 874}% \special{pa 2346 872}% \special{pa 2350 870}% \special{pa 2356 868}% \special{pa 2360 866}% \special{pa 2366 864}% \special{pa 2370 862}% \special{pa 2376 860}% \special{pa 2380 858}% \special{pa 2386 856}% \special{pa 2390 852}% \special{pa 2396 850}% \special{pa 2400 848}% \special{pa 2406 846}% \special{pa 2410 844}% \special{pa 2416 842}% \special{pa 2420 840}% \special{pa 2426 838}% \special{pa 2430 836}% \special{pa 2436 834}% \special{pa 2440 832}% \special{pa 2446 830}% \special{pa 2450 828}% \special{pa 2456 826}% \special{pa 2460 824}% \special{pa 2466 822}% \special{pa 2470 820}% \special{sp}% % CIRCLE 2 0 3 0 % 4 918 1307 918 1674 918 1674 918 1674 % \special{pn 8}% \special{ar 918 1308 368 368 0.0000000 6.2831853}% % DOT 0 0 3 0 % 1 1178 1566 % \special{pn 20}% \special{sh 1}% \special{ar 1178 1566 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 918 1307 % \special{pn 20}% \special{sh 1}% \special{ar 918 1308 10 10 0 6.28318530717959E+0000}% % LINE 2 0 3 0 % 2 918 1307 1178 1566 % \special{pn 8}% \special{pa 918 1308}% \special{pa 1178 1566}% \special{fp}% % STR 2 0 3 0 % 3 570 1338 570 1370 2 0 % {\scriptsize$\I(0,\,3)$} \put(5.7000,-13.7000){\makebox(0,0)[lb]{{\scriptsize$\I(0,\,3)$}}}% % CIRCLE 3 0 3 0 % 4 776 1709 1074 2014 1440 1473 971 1161 % \special{pn 4}% \special{ar 776 1710 426 426 5.0542565 5.9416887}% % STR 2 0 3 0 % 3 1040 1358 1040 1390 2 0 % {\scriptsize$\sqrt{2}$} \put(10.4000,-13.9000){\makebox(0,0)[lb]{{\scriptsize$\sqrt{2}$}}}% % STR 2 0 3 0 % 3 1220 1668 1220 1700 2 0 % {\scriptsize$\R(u^2,\,Bu)$} \put(12.2000,-17.0000){\makebox(0,0)[lb]{{\scriptsize$\R(u^2,\,Bu)$}}}% % STR 2 0 3 0 % 3 1990 1117 1990 1150 2 0 % {\scriptsize$C$} \put(19.9000,-11.5000){\makebox(0,0)[lb]{{\scriptsize$C$}}}% % STR 2 0 3 0 % 3 510 1017 510 1050 2 0 % {\scriptsize$K$} \put(5.1000,-10.5000){\makebox(0,0)[lb]{{\scriptsize$K$}}}% % LINE 2 0 3 0 % 2 2225 530 515 2225 % \special{pn 8}% \special{pa 2226 530}% \special{pa 516 2226}% \special{fp}% % VECTOR 0 0 3 0 % 4 1600 1050 1855 795 1855 795 1855 795 % \special{pn 20}% \special{pa 1600 1050}% \special{pa 1856 796}% \special{fp}% \special{sh 1}% \special{pa 1856 796}% \special{pa 1794 828}% \special{pa 1818 834}% \special{pa 1822 856}% \special{pa 1856 796}% \special{fp}% \special{pa 1856 796}% \special{pa 1856 796}% \special{fp}% % STR 2 0 3 0 % 3 1190 810 1190 860 2 0 % {\scriptsize$(1,\,f'(u^2))$} \put(11.9000,-8.6000){\makebox(0,0)[lb]{{\scriptsize$(1,\,f'(u^2))$}}}% % POLYLINE 2 0 3 0 % 4 1138 1525 1096 1568 1139 1610 1139 1610 % \special{pn 8}% \special{pa 1138 1526}% \special{pa 1096 1568}% \special{pa 1140 1610}% \special{pa 1140 1610}% \special{fp}% \end{picture}% \end{minipage} \begin{gather*} 2(Bu - 3)^2 - Bu(Bu - 3) = 4 \\ v^2 - 9v + 14 = (v - 2)(v - 7) = 0 \\ \therefore \,\,\, v = 2\,\,\,または\,\,\,7 \end{gather*} $v = 2$ ならば\MARU{3}から, \begin{gather*} u^4 + (2 - 3)^2 = 2 \qquad \therefore \,\,\, u = 1 \quad(\,\because\,\,\,u > 0) \end{gather*} よって $2 = v = B \cdot 1$ から $B = 2$ を得る. $v = 7$ ならば\MARU{3}から, \begin{align*} u^4 + (7 - 3)^2 = 2 \qquad \therefore \,\,\, u^4 = -14 \end{align*} となり不適. ゆえに, \begin{align*} f(x) = \textcolor{red}{\boldsymbol{2\sqrt{\vphantom{b} x}}} \tag*{$\Ans$} \end{align*} \end{document}