{"id":1966,"date":"2019-04-04T12:11:28","date_gmt":"2019-04-04T03:11:28","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1966"},"modified":"2019-09-05T19:57:45","modified_gmt":"2019-09-05T10:57:45","slug":"calculus-concavity-and-curve-sketching","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-concavity-and-curve-sketching\/","title":{"rendered":"\ud568\uc218\uc758 \ubcfc\ub85d\uc131\uacfc \uadf8\ub798\ud504\uc758 \ubaa8\uc591"},"content":{"rendered":"

\uc774\ucc28\ud568\uc218 \\(y=ax^2 + bx +c\\)\uc758 \uadf8\ub798\ud504\ub294 \\(a > 0\\)\uc77c \ub54c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\uace0 \\(a < 0\\)\uc77c \ub54c \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4. \uc0ac\uc778\uc774\ub098 \ucf54\uc0ac\uc778\uc758 \uacbd\uc6b0\uc5d0\ub294 \\(x\\)\uc758 \uac12\uc774 \ucee4\uc9d0\uc5d0 \ub530\ub77c \ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \ubcfc\ub85d\ud55c \ubc29\ud5a5\uc774 \uc704\ucabd\uacfc \uc544\ub798\ucabd\uc73c\ub85c \ubc88\uac08\uc544\uac00\uba74\uc11c \ub098\ud0c0\ub09c\ub2e4. \uc774\uc640 \uac19\uc740 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc740 \uadf8\ub798\ud504\uc758 \ubaa8\uc591\uc744 \uad00\ucc30\ud558\uba74 \uc54c \uc218 \uc788\ub2e4. \ud558\uc9c0\ub9cc \uadf8\ub798\ud504\ub97c \uadf8\ub9ac\uc9c0 \uc54a\ub354\ub77c\ub3c4 \ub3c4\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc744 \uc870\uc0ac\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc744 \uc815\uc758\ud558\uace0 \uadf8\uc640 \uad00\ub828\ub41c \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\ubcfc\ub85d\uc131\uc758 \uc815\uc758<\/h3>\n

\ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc744 \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uba87 \uac00\uc9c0\uac00 \uc788\ub2e4. \uc5ec\uae30\uc11c\ub294 \ube44\uad50\uc801 \uc5c4\ubc00\ud55c \ubc29\ubc95\uc73c\ub85c \ubcfc\ub85d\uc131\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\n

\uc815\uc758 1. (\ud568\uc218\uc758 \ubcfc\ub85d\uc131)<\/span><\/p>\n

\\(I\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \\(0 \\le \\alpha \\le 1\\)\uc778 \uc784\uc758\uc758 \\(\\alpha\\)\uc640 \\(I\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x,\\) \\(y\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[f(\\alpha x + (1-\\alpha )y ) \\le \\alpha f(x) + (1-\\alpha )f(y)\\]
    \n\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4<\/span>(convex)\u2019 \ub610\ub294 \u2018\\(f\\)\uc758 \uadf8\ub798\ud504\ub294 \\(I\\)\uc5d0\uc11c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4<\/span>\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc815\uc758\uc5ed\uc774 \uad6c\uac04\uc778 \ud568\uc218\uac00 \uc815\uc758\uc5ed\uc5d0\uc11c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud560 \ub54c \uadf8 \ud568\uc218\ub97c \ubcfc\ub85d\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n
  2. \\((-f)\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uba74 \u2018\\(f\\)\uc758 \uadf8\ub798\ud504\ub294 \\(I\\)\uc5d0\uc11c \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4<\/span>(concave)\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    \u2018\\(f\\)\uc758 \uadf8\ub798\ud504\uac00 \\(I\\)\uc5d0\uc11c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4\u2019\ub97c \uac04\ub2e8\ud788 \u2018\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4\u2019\ub77c\uace0 \ud45c\ud604\ud558\uba70, \u2018\\(f\\)\uc758 \uadf8\ub798\ud504\uac00 \\(I\\)\uc5d0\uc11c \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4\u2019\ub97c \uac04\ub2e8\ud788 \u2018\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc624\ubaa9\ud558\ub2e4\u2019\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n

    \uc815\uc758\uc5d0 \uc758\ud558\uba74 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uac00 \\(I\\)\uc758 \uc784\uc758\uc758 \ubd80\ubd84\uad6c\uac04\uc5d0\uc11c \ubcfc\ub85d\ud55c \uac83\uc774\ub2e4. \ub610\ud55c \uc77c\ucc28\ud568\uc218\uc758 \uadf8\ub798\ud504\ub294 \uad6c\uac04\uc5d0\uc11c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud55c \ub3d9\uc2dc\uc5d0 \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4.<\/p>\n

    \\(f\\)\uc758 \uadf8\ub798\ud504\uac00 \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\((-f)\\)\uc758 \uadf8\ub798\ud504\uac00 \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud55c \uac83\uc774\ubbc0\ub85c, \ud568\uc218\uc758 \ubcfc\ub85d\uc131\uc5d0 \ub300\ud558\uc5ec \ub17c\ud560 \ub54c\uc5d0\ub294 \ubcfc\ub85d\ud568\uc218(\uadf8\ub798\ud504\uac00 \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud55c \ud568\uc218)\uc5d0 \ub300\ud574\uc11c\ub9cc \ub17c\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 1. (\ubcfc\ub85d\uc131\uc758 \uae30\ud558\ud559\uc801 \uc815\uc758)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(I\\)\uc758 \uc784\uc758\uc758 \ubd80\ubd84\uad6c\uac04 \\(\\)\uc5d0 \ub300\ud558\uc5ec, \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \ub450 \uc810 \\((c,\\,f(c))\\)\uc640 \\((d,\\,f(d))\\)\ub97c \uc774\uc740 \uc120\ubd84 \uc704\uc758 \uc784\uc758\uc758 \uc810\uc774 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc758 \uc704\ucabd(above) \ub610\ub294 \uadf8\ub798\ud504\uc758 \uc704(on)\uc5d0 \ub193\uc774\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uace0 \\(x_0 \\in \\)\ub77c\uace0 \ud558\uc790.
    \n\\[x_0 = \\alpha c+ (1-\\alpha )d\\]
    \n\uc778 \\(\\alpha \\in [0,\\,1]\\)\uc744 \ud0dd\ud558\uc790. \uadf8\ub7ec\uba74 \ub450 \uc810 \\((c,\\,f(c))\\)\uc640 \\((d,\\,f(d))\\)\ub97c \uc787\ub294 \uc120\ubd84 \\(\\ell\\)\uc758 \uae30\uc6b8\uae30\ub294
    \n\\[\\frac{f(d)-f(c)}{d-c}\\]
    \n\uc774\ub2e4. \ub530\ub77c\uc11c \\((x_0 ,\\, y_0 )\\)\uc774 \uc120\ubd84 \\(\\ell\\) \uc704\uc758 \uc810\uc774\uba74
    \n\\[y_0 = \\alpha f(c) + (1-\\alpha )f(d)\\]
    \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\(f\\)\uac00 \ubcfc\ub85d\ud568\uc218\uc774\ubbc0\ub85c \\(f(x_0 )\\le y_0\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \uc774 \uacfc\uc815\uc744 \uac70\uc2ac\ub7ec \uc62c\ub77c\uac00\uba74 \uc5ed\uc774 \uc99d\uba85\ub41c\ub2e4.<\/p>\n

    <\/span><\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 2. (\uae30\uc6b8\uae30\ub97c \uc774\uc6a9\ud55c \ud568\uc218\uc758 \ubcfc\ub85d\uc131\uc758 \uc815\uc758)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubcfc\ub85d\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\((a,\\,b)\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uae30\uc6b8\uae30\uac00 \uc99d\uac00\ud568\uc218\uc778 \uac83\uc774\ub2e4. \uc989
    \n\\[(a < c < x < d < b) \\quad \\Rightarrow \\quad \\frac{f(x)-f(c)}{x-c} \\le \\frac{f(d)-f(x)}{d-x}\\]\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(a < c < x < d < b\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ub450 \uc810 \\((c,\\,f(c)),\\) \\((d,\\,f(d))\\)\ub97c \uc787\ub294 \uc9c1\uc120\uc744 \uadf8\ub798\ud504\ub85c \uac16\ub294 \uc77c\ucc28\ud568\uc218\ub97c \\(\\lambda (x)\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \ubcfc\ub85d\ud558\uba74 \\(f(x) \\le \\lambda (x)\\)\uc774\ubbc0\ub85c\n\\[\\frac{f(x)-f(c)}{x-c} \\le \\frac{\\lambda (x)-\\lambda (c)}{x-c} = \\frac{\\lambda (d) - \\lambda (x)}{d-x} \\le \\frac{f(d) - f(x)}{d-x}\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc5ed\uc73c\ub85c \\(f\\)\uac00 \ubcfc\ub85d\ud568\uc218\uac00 \uc544\ub2c8\uba74 \uc801\ub2f9\ud55c \\(x\\in (c,\\,d)\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lambda (x) < f(x)\\)\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\\frac{f(x)-f(c)}{x-c} \\ge \\frac{\\lambda (x) - \\lambda (c)}{x-c} = \\frac{\\lambda (d) - \\lambda (x)}{d-x} > \\frac{f(d) – f(x)}{d-x}\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc740 \\((a,\\,b)\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uae30\uc6b8\uae30\uac00 \uc99d\uac00\ud558\uc9c0 \uc54a\ub294 \ubd80\ubd84\uc774 \uc874\uc7ac\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ud568\uc218\uc758 \ubcfc\ub85d\uc131\uc740 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uae30\uc6b8\uae30\uc640 \uad00\ub828\uc788\uc73c\ubbc0\ub85c, \ub3c4\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ubcfc\ub85d\uc131\uc744 \ud310\uc815\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 3. (\ubcfc\ub85d\uc131\uc758 \ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

    \\(f\\)\uac00 \uc5f4\ub9b0\uad6c\uac04 \\(I = (a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(I\\)\uc5d0\uc11c \\(f ‘ \\)\uc774 \uc99d\uac00\ud568\uc218\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \\(c,\\) \\(d\\)\uac00 \\(I\\)\uc758 \ub450 \uc810\uc774\uba70 \\(c < d\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ucda9\ubd84\ud788 \uc791\uc740 \uc591\uc218 \\(h\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(c+h < d\\)\uc640 \\(d+h < b\\)\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec\n\\[\\frac{f(c+h) - f(c)}{h} \\le \\frac{f(d+h) - f(d)}{h}\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc591\ubcc0\uc5d0 \\(h \\to 0\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(f ' (c) \\le f ' (d)\\)\ub97c \uc5bb\ub294\ub2e4. \uc989 \\(f ' \\)\uc740 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ub2e4.<\/p>\n

    \ub2e4\uc74c\uc73c\ub85c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(f ‘ \\)\uc774 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(a < c < x < d < b\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec, \\(c\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \\(x_0\\)\uc774 \uc874\uc7ac\ud558\uc5ec\n\\[\\frac{f(x)-f(c)}{x-c} = f ' (x_0 )\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0, \\(x\\)\uc640 \\(d\\) \uc0ac\uc774\uc5d0 \\(x_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec\n\\[\\frac{f(d)-f(x)}{d-x} = f ' (x_1 )\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\(x_0 < x_1\\)\uc774\ubbc0\ub85c \\(f ' (x_0 )\\le f ' (x_1 )\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \n

    \ub530\ub984\uc815\ub9ac 1.<\/span><\/p>\n

    \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(I\\)\uc5d0\uc11c \\(f ‘ ‘ \\ge 0\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(f ‘ ‘ (x) = \\frac{d}{dx} f ‘ (x)\\)\uc774\ubbc0\ub85c \\(I\\)\uc5d0\uc11c \\(f ‘ ‘ \\ge 0\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f ‘ \\)\uc774 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc778 \uac83\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec \\(I\\)\uc5d0\uc11c \\(f ‘ ‘ \\ge 0\\)\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc778 \uac83\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \\(f\\)\uac00 \uad6c\uac04\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc815\ub9ac 1, 2, 3\uacfc \ub530\ub984\uc815\ub9ac 1\uc740 \ubaa8\ub450 \uc815\uc758 1\uacfc \ub3d9\uce58\uc778 \uc815\uc758\ub97c \uc81c\uacf5\ud55c\ub2e4. \uc989 \\(D^2\\)\uae09 \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \ub2e4\uc12f \uac1c\uc758 \uba85\uc81c \uc911 \uc5b4\ub290 \uac83\uc774\ub4e0 \ubcfc\ub85d\uc131\uc758 \uc815\uc758\ub85c \uc0ac\uc6a9\ud574\ub3c4 \ub41c\ub2e4.<\/p>\n

    <\/p>\n

    \ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \ubcfc\ub85d\ud568\uc218\uc758 \uc131\uc9c8<\/h3>\n

    \ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \ubcfc\ub85d\ud568\uc218\uc758 \uc5ec\ub7ec \uac00\uc9c0 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

    \n

    \uc815\ub9ac 4. (\ubcfc\ub85d\ud568\uc218\uc758 \uc5f0\uc18d\uc131)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\(I = (a,\\,b)\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/p>\n

    \\(x_0 \\in I\\)\ub77c\uace0 \ud558\uc790. \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \uc6b0\uadf9\ud55c\uc774 \\(f(x_0 )\\)\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n

    \\(a < c < x_0 < x < d < b\\)\ub77c\uace0 \ud558\uc790. \ub450 \uc810 \\((c,\\,f(c))\\)\uc640 \\((x_0 ,\\,f(x_0 ))\\)\uc744 \uc787\ub294 \uc9c1\uc120\uc744 \uadf8\ub798\ud504\ub85c \uac16\ub294 \uc77c\ucc28\ud568\uc218\ub97c \\(g\\)\ub77c\uace0 \ud558\uace0, \ub450 \uc810 \\((x_0 ,\\, f(x_0 ))\\)\uacfc \\((d,\\,f(d))\\)\ub97c \uc787\ub294 \uc9c1\uc120\uc744 \uadf8\ub798\ud504\ub85c \uac16\ub294 \uc77c\ucc28\ud568\uc218\ub97c \\(h\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \ubcfc\ub85d\ud568\uc218\uc774\ubbc0\ub85c \uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(f(x) \\le h(x)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc810 \\((x_0 ,\\,f(x_0 ))\\)\uc740 \\((c,\\,f(c))\\)\uc640 \\((x,\\,f(x))\\)\ub97c \uc787\ub294 \uc120\ubd84 \uc704(on)\uc5d0 \uc788\uac70\ub098 \ub610\ub294 \uadf8 \uc544\ub798\ucabd(below)\uc5d0 \uc788\uc73c\ubbc0\ub85c \\(g(x) \\le f(x)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc784\uc758\uc758 \\(x\\in (x_0 ,\\,d)\\)\uc5d0 \ub300\ud558\uc5ec\n\\[g(x) \\le f(x) \\le h(x)\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \\(y=g(x)\\)\uc640 \\(y=h(x)\\)\uc758 \uadf8\ub798\ud504\uac00 \uc810 \\((x_0 ,\\, f(x_0 ))\\)\uc744 \uc9c0\ub098\ubbc0\ub85c \\(x \\to x_0 ^+\\)\uc77c \ub54c \\(g(x) \\to g(x_0 ) ,\\) \\(h(x) \\to f(x_0 )\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc0cc\ub4dc\uc704\uce58 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(f(x) \\to f(x_0 )\\)\uc744 \uc5bb\ub294\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \uc88c\uadf9\ud55c\ub3c4 \\(f(x_0 )\\)\uc5d0 \uc218\ub834\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\(x_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n

    \\(x_0\\)\uc774 \\(I\\)\uc758 \uc784\uc758\uc758 \uc810\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \n

    \ucc38\uace0.<\/span>
    \n\\(I\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04\uc774 \uc544\ub2c8\uba74 \uc815\ub9ac 4\ub294 \uc131\ub9bd\ud558\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(I = [0,\\,1]\\)\uc5d0\uc11c
    \n\\[f(x) =
    \n\\begin{cases}
    \n0 &\\quad\\text{if} \\,\\, 0 \\le x < 1 \\\\[8pt]\n1 &\\quad\\text{if} \\,\\, 1 \\le x=1\n\\end{cases}\\]\n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uc9c0\ub9cc \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4.\n<\/span><\/p>\n<\/div>\n

    \ud568\uc218 \\(f\\)\uac00 \\(x_0\\)\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( \\lvert x-x_0 \\rvert < \\delta\\)\uc77c \ub54c\ub9c8\ub2e4 \\(f(x) < f(x_0 )\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(x_0\\)\uc5d0\uc11c \uace0\uc720\uadf9\ub313\uac12<\/span>(proper maximum)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \uc774 \ubd80\ub4f1\uc2dd\uc774 \ubc18\ub300\ub85c \uc131\ub9bd\ud558\uba74, \uc989 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\( \\lvert x-x_0 \\rvert < \\delta\\)\uc77c \ub54c\ub9c8\ub2e4 \\(f(x) > f(x_0 )\\)\uc774 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(x_0\\)\uc5d0\uc11c \uace0\uc720\uadf9\uc19f\uac12<\/span>(proper minimum)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 5. (\ubcfc\ub85d\ud568\uc218\uc758 \uace0\uc720\uadf9\uac12)<\/span><\/p>\n

      \n
    1. \ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uba74 \\(f\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \uace0\uc720\uadf9\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n
    2. \ud568\uc218 \\(f\\)\uac00 \\([0,\\,\\infty )\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uace0 \uace0\uc720\uadf9\uc19f\uac12\uc744 \uac00\uc9c0\uba74 \\(x\\to \\infty\\)\uc77c \ub54c \\(f(x) \\to \\infty\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
      \n

      \uc99d\uba85<\/p>\n

      [1] \uacb0\ub860\uc5d0 \ubc18\ud558\uc5ec \\(f\\)\uac00 \\(x_0 \\in (a,\\,b)\\)\uc5d0\uc11c \uace0\uc720\uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(c < x_0 < d\\)\uc778 \\(c,\\) \\(d\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(c < x < d\\)\uc77c \ub54c\ub9c8\ub2e4 \\(f(x) < f(x_0 )\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ud2b9\ud788 \ub450 \uc810 \\((c,\\,f(c))\\)\uc640 \\((d,\\,f(d))\\)\ub97c \uc787\ub294 \uc120\ubd84 \uc704\uc758 \uc810\uc758 \\(y\\)\uc88c\ud45c\ub294 \\(f(x_0 )\\)\ubcf4\ub2e4 \uc791\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(f\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \uace0\uc720\uadf9\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

      [2] \\(f\\)\uac00 \\(x_0 \\in (a,\\,b)\\)\uc5d0\uc11c \uace0\uc720\uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \\(x_1 > x_0\\)\uc778 \uc810 \\(x_1\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ub450 \uc810 \\((x_0 ,\\, f(x_0 ))\\)\uacfc \\((x_1 ,\\, f(x_1 ))\\)\uc744 \uc787\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \\(y=g(x)\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \uace0\uc720\uadf9\uc19f\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c \\(f(x_1 ) > f(x_0) \\)\uacfc \\(x_1 > x_0\\)\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(x_1\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub77c\uc11c \\(g\\)\uc758 \uadf8\ub798\ud504\uc758 \uae30\uc6b8\uae30\ub294 \uc591\uc218\uc774\ub2e4. \ub354\uc6b1\uc774 \uc815\ub9ac 4\uc758 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \uc0b4\ud3b4\ubcf8 \ubc14\uc640 \uac19\uc774 \uc784\uc758\uc758 \\(x\\in (x_1 ,\\, \\infty )\\)\uc5d0 \ub300\ud558\uc5ec \\(g(x) \\le f(x)\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(x\\to \\infty\\)\uc77c \ub54c \\(g(x) \\to \\infty\\)\uc774\ubbc0\ub85c \\(f(x) \\to \\infty\\)\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 6. (\ubcfc\ub85d\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uba74 \\((a,\\,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \uc88c\ubbf8\ubd84\uacc4\uc218\uc640 \uc6b0\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\uba70 \uadf8\ub4e4\uc740 \uac01\uac01 \\((a,\\,b)\\)\uc5d0\uc11c \uc720\ud55c\uac12\uc744 \uac16\uace0 \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(h < 0\\)\uc774\uba74 \ub450 \uc810 \\((x,\\,f(x))\\)\uc640 \\((x+h ,\\,f(x+h))\\)\ub97c \uc787\ub294 \uc9c1\uc120\uc758 \uae30\uc6b8\uae30\ub294\n\\[\\frac{f(x+h)-f(x)}{h}\\]\n\uc774\ub2e4. \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \uc74c\uc218 \\(h\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6cc\uc9c8\uc218\ub85d \uc774 \ubd84\uc218\uc2dd\uc758 \uac12\uc740 \uc99d\uac00\ud55c\ub2e4. \ub530\ub77c\uc11c \ub2e8\uc870\uc218\ub834 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(h\\to 0^-\\)\uc77c \ub54c \uc774 \ubd84\uc218\uc2dd\uc740 \uc720\ud55c\uac12\uc5d0 \uc218\ub834\ud55c\ub2e4. \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \\(h\\to 0^+\\)\uc77c \ub54c\uc5d0\ub3c4 \uc774 \ubd84\uc218\uc2dd\uc758 \uac12\uc774 \uc720\ud55c\uac12\uc5d0 \uc218\ub834\ud568\uc744 \uc54c \uc218 \uc788\ub2e4. \uc989 \\(f_l ' (x) \\le f_r ' (x)\\)\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

      \uc774\uc81c \\(f_r ‘ (x)\\)\uac00 \\((a,\\,b)\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc784\uc744 \ubcf4\uc774\uc790. \\(x_1 ,\\) \\(u,\\) \\(t,\\) \\(x_2\\)\uac00 \\((a,\\,b)\\)\uc758 \uc810\uc774\uace0 \\(x_1 < u < t < x_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\n\\[\\frac{f(u)-f(x_1 )}{u-x_1} \\le \\frac{f(x_2 ) - f(t)}{x_2 -t}\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc704 \ubd80\ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc5d0 \\(u \\to x_1^+\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uace0 \uc6b0\ubcc0\uc5d0 \\(t\\to x_2^-\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74\n\\[f_r ' (x_1) \\le f_l ' (x_2 ) \\]\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7f0\ub370 \\(f_l ' (x_2) \\le f_r ' (x_2)\\)\uc774\ubbc0\ub85c\n\\[f_r ' (x_1) \\le f_l ' (x_2) \\le f_r ' (x_2)\\]\n\uc774\ub2e4. \uc989 \\(f_r ' (x)\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \uc99d\uac00\ud55c\ub2e4. \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(f_l ' (x)\\)\ub3c4 \\((a,\\,b)\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.\n<\/span><\/p>\n<\/div>\n

      \n

      \ub530\ub984\uc815\ub9ac 2. (\ubcfc\ub85d\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\uba74 \\((a,\\,b)\\)\uc758 \uc810 \uc911\uc5d0\uc11c \\(f\\)\uac00 \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud55c \uac83\ub4e4\uc758 \ubaa8\uc784\uc740 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc774\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\((a,\\,b)\\)\uc758 \uc810 \uc911\uc5d0\uc11c \uadf8 \uc810\uc5d0\uc11c \\(f\\)\uc758 \uc88c\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uac70\ub098 \uc6b0\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc740 \uac83\ub4e4\uc758 \ubaa8\uc784\uc744 \\(E\\)\ub77c\uace0 \ud558\uc790. \uad6c\uac04\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc758 \ubd88\uc5f0\uc18d\uc810\uc758 \ubaa8\uc784\uc740 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc778\ub370, \\(f_l ‘ \\)\uacfc \\(f_r ‘ \\)\uc740 \ubaa8\ub450 \\((a,\\,b)\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ubbc0\ub85c \\(E\\)\ub294 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc774\ub2e4.<\/p>\n

      \\(x_0 \\in (a,\\,b) \\setminus E\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc815\ub9ac 6\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
      \n\\[f_r ‘ (x) \\le f_l ‘ (x_0 ) \\le f_r ‘ (x_0 ).\\]
      \n\\(f_r ‘\\)\uacfc \\(f_l ‘\\)\uc774 \ubaa8\ub450 \\(x_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uc704 \uc2dd\uc5d0 \\(x \\to x_0\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74
      \n\\[f_r ‘ (x_0 ) \\le f_l ‘ (x_0 ) \\le f_r ‘ (x_0 )\\]
      \n\uc744 \uc5bb\ub294\ub2e4. \uc989 \\(f_r ‘ (x_0 ) = f_l ‘ (x_0 )\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud55c \uc810\uc740 \ubaa8\ub450 \\(E\\)\uc5d0\ub9cc \uc874\uc7ac\ud55c\ub2e4. \\(E\\)\uac00 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc774\uace0, \uac00\uc0b0\uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc740 \uac00\uc0b0\uc9d1\ud569\uc774\ubbc0\ub85c \uc815\ub9ac\uc758 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \ub530\ub984\uc815\ub9ac 3. (\ud568\uc218\uc758 \uc99d\uac10\uc5d0 \ub300\ud55c \ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uc9d1\ud569
      \n\\[E = \\left\\{ x \\in (a,\\,b) \\,\\vert\\, f ‘ (x) < 0 \\right\\}\\]\n\uc774 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uace0 \\(x_1 \\in E\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(y\\in (f ‘ (x_1 ) ,\\, 0)\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\ub974\ubd80\uc758 \uc815\ub9ac(\ub3c4\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac)\uc5d0 \uc758\ud558\uc5ec \\(x \\in (a,\\,b)\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f ‘ (x) = y < 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc989 \\((f ' (x_1 ) ,\\, 0)\\)\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uba70, \uc774 \uc9d1\ud569\uc5d0 \uc18d\ud55c \ubaa8\ub4e0 \\(y\\)\uc5d0 \ub300\ud558\uc5ec \\(f ' (x) = y\\)\uc778 \\(x\\)\uac00 \\(E\\)\uc5d0 \uc874\uc7ac\ud558\uac8c \ub418\ubbc0\ub85c \\(E\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\ub2e4. \uc774\uac83\uc740 \ubaa8\uc21c\uc774\ubbc0\ub85c \\(E\\)\ub294 \uacf5\uc9d1\ud569\uc774\ub2e4.<\/p>\n

      \uadf8\ub7ec\ubbc0\ub85c \\((a,\\,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f ‘ \\ge 0\\)\uc774\ub2e4. \uc989 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc99d\uac00\ud568\uc218\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \ub530\ub984\uc815\ub9ac 4. (\uc0c1\uc218\ud568\uc218\uc758 \ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \uc9d1\ud569
      \n\\[E = \\left\\{ x \\in (a,\\,b) \\,\\vert\\, f ‘ (x) \\ne 0 \\right\\}\\]
      \n\uc774 \uc720\ud55c\uc774\uac70\ub098 \uac00\uc0b0\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(f\\)\uc640 \\((-f)\\)\uc5d0 \ub530\ub984\uc815\ub9ac 3\uc744 \uc801\uc6a9\ud558\uba74 \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      <\/p>\n

      \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubaa8\uc591<\/h3>\n

      \ub530\ub984\uc815\ub9ac 1\uc744 \ud655\uc7a5\ud558\uba74 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 7. (\ubcfc\ub85d\uc131\uc758 \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

      \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f ‘ ‘ > 0\\)\uc774\uba74 \\(f\\)\uc758 \uadf8\ub798\ud504\ub294 \\(I\\)\uc5d0\uc11c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4.<\/li>\n
      2. \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f ‘ ‘ < 0\\)\uc774\uba74 \\(f\\)\uc758 \uadf8\ub798\ud504\ub294 \\(I\\)\uc5d0\uc11c \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

        \\(x\\)\uc758 \uac12\uc774 \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \ud568\uc218 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uac00 \ubcfc\ub85d\ud55c \ubc29\ud5a5\uc774 \ubc14\ub014 \ub54c \uc810 \\((c,\\,f(c))\\)\ub97c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ubcc0\uace1\uc810<\/span>(point of inflection)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (\uc774 \uc815\uc758\ub294 \ucc45\ub9c8\ub2e4 \uc870\uae08\uc529 \ub2e4\ub974\ub2e4. \uc989 \ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uc870\uac74\uc774 \ucd94\uac00\ub418\uae30\ub3c4 \ud55c\ub2e4.) \ubcc0\uace1\uc810\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

        \n

        \\((c,\\,f(c))\\)\uac00 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc758 \ubcc0\uace1\uc810\uc774\uba74 \\(f ‘ ‘ (c)=0\\)\uc774\uac70\ub098 \ub610\ub294 \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uacc4\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n

        \ub3c4\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf9\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4 \ubcf4\uc790.<\/p>\n

        \n

        \uc815\ub9ac 8. (\uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \uc815\uc758\uc5ed\uc758 \ud55c \ubd80\ubd84\uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\uba70, \\(f\\)\uac00 \\(c\\)\ub97c \uc81c\uc678\ud55c \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. (\\(c\\)\uc5d0\uc11c\ub294 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud560 \uc218\ub3c4 \uc788\uace0 \uadf8\ub807\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4.)<\/p>\n

          \n
        1. \\(x\\)\uc758 \uac12\uc774 \uc99d\uac00\ud558\uba74\uc11c \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f ‘ \\)\uc758 \uac12\uc774 \uc74c\uc218\uc5d0\uc11c \uc591\uc218\ub85c \ubc14\ub00c\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
        2. \\(x\\)\uc758 \uac12\uc774 \uc99d\uac00\ud558\uba74\uc11c \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f ‘ \\)\uc758 \uac12\uc774 \uc591\uc218\uc5d0\uc11c \uc74c\uc218\ub85c \ubc14\ub00c\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
        3. \\(x\\)\uc758 \uac12\uc774 \uc99d\uac00\ud558\uba74\uc11c \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f ‘ \\)\uc758 \uac12\uc758 \ubd80\ud638\uac00 \ubc14\ub00c\uc9c0 \uc54a\uc73c\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \uc99d\uba85<\/p>\n

          \uc5ec\uae30\uc11c\ub294 [1]\ub9cc \uc99d\uba85\ud55c\ub2e4. \\(x < c\\)\uc77c \ub54c \\(f ' (x) < 0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(c\\)\uc758 \uc67c\ucabd\uc5d0\uc11c \uac10\uc18c\ud55c\ub2e4. \ub530\ub77c\uc11c \\(x < c\\)\uc778 \\(x\\in I\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(f(x) > f(c)\\)\uc774\ub2e4. \ub610\ud55c \\(c < x\\)\uc77c \ub54c \\(f ' (x) > 0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(c\\)\uc758 \uc624\ub978\ucabd\uc5d0\uc11c \uc99d\uac00\ud55c\ub2e4. \ub530\ub77c\uc11c \\(c < x\\)\uc778 \\(x\\in I\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(f(c) < f(x)\\)\uc774\ub2e4. \uc989 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud574\uc11c \\(f(x) \\ge f(c)\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

          [2]\uc640 [3]\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ub41c\ub2e4.
          \n<\/span><\/p>\n<\/div>\n

          \n

          \uc815\ub9ac 9. (\uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

          \ud568\uc218 \\(f\\)\uac00 \uc815\uc758\uc5ed\uc758 \ud55c \ubd80\ubd84\uad6c\uac04 \\(I\\)\uc5d0\uc11c \\(C^2\\)\uae09\uc774\uace0(\uc5f0\uc18d\uc778 \uc774\uacc4\ub3c4\ud568\uc218\ub97c \uac16\uace0) \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

            \n
          1. \ub9cc\uc57d \\(f ‘ (c) = 0\\)\uc774\uace0 \\(f ‘ ‘ (c) < 0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
          2. \ub9cc\uc57d \\(f ‘ (c) = 0\\)\uc774\uace0 \\(f ‘ ‘ (c) > 0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
          3. \ub9cc\uc57d \\(f ‘ (c) = 0\\)\uc774\uace0 \\(f ‘ ‘ (c) = 0\\)\uc774\uba74 \uacb0\ub860\uc744 \ub0b4\ub9b4 \uc218 \uc5c6\ub2e4. \uc989 \uc774 \uacbd\uc6b0 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c8 \uc218\ub3c4 \uc788\uace0 \uadf9\uc19f\uac12\uc744 \uac00\uc9c8 \uc218\ub3c4 \uc788\uc73c\uba70 \uadf9\uac12\uc744 \uac16\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
            \n

            \uc99d\uba85<\/p>\n

            [1] \\(f ‘ ‘ (c) < 0\\)\uc774\uace0 \\(f ' ' \\)\uc774 \uc5f0\uc18d\uc774\ubbc0\ub85c \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\(J\\) \uc704\uc5d0\uc11c \\(f ' ' < 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f ' \\)\uc740 \\(J\\)\uc5d0\uc11c \uac10\uc18c\ud568\uc218\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f ' (c)=0\\)\uc774\ubbc0\ub85c \\(x\\)\uc758 \uac12\uc774 \uc99d\uac00\ud558\uba74\uc11c \\(c\\)\ub97c \uc9c0\ub098\ub294 \uc21c\uac04 \\(f ' \\)\uc758 \uac12\uc740 \uc591\uc218\uc5d0\uc11c \uc74c\uc218\ub85c \ubc14\ub010\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

            [2]\ub3c4 [1]\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ub41c\ub2e4.<\/p>\n

            [3]\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc138 \uac00\uc9c0 \uc608\ub97c \ucc3e\uc73c\uba74 \ub41c\ub2e4. \uba3c\uc800 \\(f_1 (x) = x^4\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(f_1 ‘ (0) = 0,\\) \\(f_1 ‘ ‘ (0) = 0\\)\uc774\uc9c0\ub9cc \\(f_1\\)\uc740 \\(0\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(f_2 (x) = -x^4\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(f_2 ‘ (0) = 0,\\) \\(f_2 ‘ ‘ (0) = 0\\)\uc774\uc9c0\ub9cc \\(f_2\\)\ub294 \\(0\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4. \ub05d\uc73c\ub85c \\(f_3 (x) = x^3\\)\uc5d0 \ub300\ud574\uc11c\ub294 \\(f_3 ‘ (0) = 0,\\) \\(f_3 ‘ ‘ (0) = 0\\)\uc774\uc9c0\ub9cc \\(f_3\\)\uc740 \\(0\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            \uc60c\uc13c\uc758 \ubd80\ub4f1\uc2dd<\/h3>\n

            \uc774 \ud3ec\uc2a4\ud2b8\ub294 \ubbf8\uc801\ubd84\ud559 \ubbf8\ubd84 \ub2e8\uc6d0\uc5d0 \uc18d\ud558\ub294 \uae00\uc774\uae30 \ub54c\ubb38\uc5d0 \uc9c0\uae08\uae4c\uc9c0 \ubbf8\ubd84\uacfc \uad00\ub828\ub41c \ub0b4\uc6a9\ub9cc \uc11c\uc220\ud558\uc600\ub2e4. \ube44\ub85d \uadf9\ud55c\uacfc \ubbf8\ubd84 \ub2e8\uc6d0\uc758 \ubc94\uc704\uc5d0\ub294 \ubc97\uc5b4\ub098\uc9c0\ub9cc \ubcfc\ub85d\ud568\uc218\uc758 \uc131\uc9c8\uc744 \ub2e4\ub8e8\uba74\uc11c \uc0b4\ud3b4\ubcf4\uc9c0 \uc54a\uace0 \ub118\uc5b4\uac00\uae30\uc5d0\ub294 \ucc38 \uc544\uc26c\uc6b4 \uac83\uc774 \ud558\ub098 \uc788\ub294\ub370, \uadf8\uac83\uc740 \ubc14\ub85c \uc60c\uc13c\uc758 \ubd80\ub4f1\uc2dd<\/span>(Jensen\u2019s inequality)\uc774\ub2e4.<\/p>\n

            \n

            \uc815\ub9ac 10. (\uc60c\uc13c\uc758 \ubd80\ub4f1\uc2dd)<\/span><\/p>\n

            \\(\\phi\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc774\uace0 \ud568\uc218 \\(f : [ 0,\\,1] \\to [a,\\,b]\\)\uc640 \\(\\phi \\circ f\\)\uac00 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c
            \n\\[\\phi \\left( \\int_{a}^{b} f(x) dx \\right) \\le \\int_0^1 ( \\phi \\circ f )(x) dx \\tag{1}\\]
            \n\uac00 \uc131\ub9bd\ud55c\ub2e4. <\/p>\n<\/div>\n

            \n

            \uc99d\uba85<\/p>\n

            \uc784\uc758\uc758 \\(x\\in [0,\\,1]\\)\uc5d0 \ub300\ud558\uc5ec \\(a \\le f(x) \\le b\\)\uc774\ubbc0\ub85c
            \n\\
            \n\ub77c\uace0 \ud558\uba74 \\(c \\in [a,\\,b]\\)\uc774\ub2e4. \ub610\ud55c
            \n\\[s := \\sup \\left\\{ \\left. \\frac{\\phi (c) – \\phi (x)}{c-x} \\right\\vert x \\in [a,\\,c) \\right\\}\\]
            \n\ub77c\uace0 \ud558\uc790. \uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(u \\in (c,\\,b]\\)\uc640 \\(x\\in [a,\\,c)\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\frac{\\phi (c)-\\phi (x)}{c-x} \\le \\frac{\\phi (u) – \\phi (c)}{u-c}\\tag{2}\\]
            \n\uc774\ubbc0\ub85c \\(s\\)\ub294 \uc798 \uc815\uc758\ub41c \uac12\uc774\ub2e4. \ub530\ub77c\uc11c \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec
            \n\\[s \\le \\frac{\\phi (u) – \\phi (c)}{u-c}\\]
            \n\uc774\uba70, \uc774 \ubd80\ub4f1\uc2dd\uc73c\ub85c\ubd80\ud130 \uc784\uc758\uc758 \\(u\\in \\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
            \n\\[\\phi(c) + s(u-c) \\le \\phi (u).\\tag{3}\\]
            \n\ud55c\ud3b8 \\(s\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec(\uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec) \\(u\\in [a,\\,c)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
            \n\\[s \\ge \\frac{\\phi (c) – \\phi (u)}{c-u}.\\]
            \n\ub530\ub77c\uc11c \uc784\uc758\uc758 \\(u\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec (3)\uc774 \uc131\ub9bd\ud55c\ub2e4. (3)\uc5d0\uc11c \\(u=f(x)\\)\ub77c\uace0 \ud558\uba74
            \n\\[\\phi (c) + s(f(x)-c) \\le (\\phi \\circ f)(x)\\]
            \n\ub97c \uc5bb\ub294\ub2e4. \\([0,\\,1]\\) \uc704\uc5d0\uc11c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc704 \ubd80\ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \uc801\ubd84\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
            \n\\[\\phi (c) + s \\left( \\int_0^1 f(x) dx – c \\right) \\le \\int_0^1 (\\phi \\circ f)(x) dx.\\tag{4}\\]
            \n\ud55c\ud3b8 \uc784\uc758\uc758 \uc2e4\uc218 \\(s\\)\uc5d0 \ub300\ud558\uc5ec
            \n\\[\\phi \\left( \\int_0^1 f(x) dx \\right) = \\phi (c) + s \\left( \\int_0^1 f(x) dx -c \\right)\\tag{5}\\]
            \n\uac00 \uc131\ub9bd\ud55c\ub2e4. (4)\uc640 (5)\ub97c \uacb0\ud569\ud558\uba74 (1)\uc744 \uc5bb\ub294\ub2e4.
            \n<\/span><\/p>\n<\/div>\n

            <\/p>\n

            <\/p>\n","protected":false},"excerpt":{"rendered":"

            \uc774\ucc28\ud568\uc218 \\(y=ax^2 + bx +c\\)\uc758 \uadf8\ub798\ud504\ub294 \\(a > 0\\)\uc77c \ub54c \uc544\ub798\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\uace0 \\(a < 0\\)\uc77c \ub54c \uc704\ucabd\uc73c\ub85c \ubcfc\ub85d\ud558\ub2e4. \uc0ac\uc778\uc774\ub098 \ucf54\uc0ac\uc778\uc758 \uacbd\uc6b0\uc5d0\ub294 \\(x\\)\uc758 \uac12\uc774 \ucee4\uc9d0\uc5d0 \ub530\ub77c \ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \ubcfc\ub85d\ud55c \ubc29\ud5a5\uc774 \uc704\ucabd\uacfc \uc544\ub798\ucabd\uc73c\ub85c \ubc88\uac08\uc544\uac00\uba74\uc11c \ub098\ud0c0\ub09c\ub2e4. \uc774\uc640 \uac19\uc740 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc740 \uadf8\ub798\ud504\uc758 \ubaa8\uc591\uc744 \uad00\ucc30\ud558\uba74 \uc54c \uc218 \uc788\ub2e4. \ud558\uc9c0\ub9cc \uadf8\ub798\ud504\ub97c \uadf8\ub9ac\uc9c0 \uc54a\ub354\ub77c\ub3c4 \ub3c4\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc744 \uc870\uc0ac\ud560 \uc218 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubcfc\ub85d\uc131\uc744 \uc815\uc758\ud558\uace0 \uadf8\uc640 \uad00\ub828\ub41c \uc131\uc9c8\uc744…\n<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[257,140,258,263,261,260,255,256,262,259],"class_list":["post-1966","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-concavity","tag-convergence","tag-convex","tag-jensens-inequality","tag-point-of-inflection","tag-260","tag-255","tag-256","tag-262","tag-259"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1966","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=1966"}],"version-history":[{"count":27,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1966\/revisions"}],"predecessor-version":[{"id":3226,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1966\/revisions\/3226"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}