{"id":1962,"date":"2019-04-02T12:09:21","date_gmt":"2019-04-02T03:09:21","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1962"},"modified":"2019-09-05T19:56:28","modified_gmt":"2019-09-05T10:56:28","slug":"calculus-the-mean-value-theorem","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-the-mean-value-theorem\/","title":{"rendered":"\ud3c9\uade0\uac12 \uc815\ub9ac"},"content":{"rendered":"<p>\ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \ub0b4\uc810 \\(a\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\uacc4\uc218 \\(f &#8216; (a)\\)\ub294 \\(x=a\\)\uc77c \ub54c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \uae30\uc6b8\uae30\uc640 \uac19\ub2e4. \uc989 \ubbf8\ubd84\uacc4\uc218\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubaa8\uc591\uc5d0 \uc758\ud558\uc5ec \uacb0\uc815\ub418\ubbc0\ub85c, \ubbf8\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\uac12, \uc99d\uac00\uc640 \uac10\uc18c\ub97c \uc815\uc758\ud558\uace0 \uc774\uc640 \uad00\ub828\ub41c \ubbf8\ubd84\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\ud568\uc218\uc758 \uadf9\uac12<\/h3>\n<p>\\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(M \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\le M\\)\uc774\uba74 \\(M\\)\uc744 \\(E\\)\uc758 \ucd5c\ub313\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(m \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(m\\le x\\)\uc774\uba74 \\(m\\)\uc744 \\(E\\)\uc758 \ucd5c\uc19f\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ud568\uc218\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 1. (\ud568\uc218\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12)<\/span><\/p>\n<p>\\(f\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(D\\)\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(c_1 \\in D\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le f(c_1 )\\)\uc774\uba74 \u2018\\(f\\)\ub294 \\(c_1\\)\uc5d0\uc11c <span class=\"defined\">\ucd5c\ub313\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019 \ub610\ub294 \u2018\\(f\\)\ub294 \\(D\\) \uc704\uc5d0\uc11c <span class=\"defined\">\ucd5c\ub313\uac12<\/span> \\(f(c_1)\\)\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(c_2 \\in D\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in D\\)\uc5d0 \ub300\ud558\uc5ec \\(f(c_2 ) \\le f(x )\\)\uc774\uba74 \u2018\\(f\\)\ub294 \\(c_2\\)\uc5d0\uc11c <span class=\"defined\">\ucd5c\uc19f\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019 \ub610\ub294 \u2018\\(f\\)\ub294 \\(D\\) \uc704\uc5d0\uc11c <span class=\"defined\">\ucd5c\uc19f\uac12<\/span> \\(f(c_2)\\)\ub97c \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uc9d1\ud569\uacfc \uc810\uc744 \ud63c\ub3d9\ud560 \uc5fc\ub824\uac00 \uc5c6\uc744 \ub54c\uc5d0\ub294 \u2018\\(f\\)\uac00 \\(D\\) \uc704\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\u2019\ub97c \uac04\ub2e8\ud788 \u2018\\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f(x)= x^2\\)\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790 \ub9cc\uc57d \uc774 \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \\(D_1 = \\mathbb{R}\\)\ub77c\uba74 \uc774 \ud568\uc218\ub294 \\(x=0\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c0\uba70, \ucd5c\ub313\uac12\uc740 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \ub9cc\uc57d \uc774 \ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \\(D_2 = [1 ,\\,2]\\)\ub77c\uba74 \uc774 \ud568\uc218\ub294 \\(x=1\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c0\uba70 \\(x=2\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<p>\uc774\ubc88\uc5d0\ub294 \uc815\uc758\uc5ed\uc774 \\(D_3 = [-1,\\,2]\\)\uc778 \ud568\uc218 \\(f(x)=x^2\\)\uc744 \uc0dd\uac01\ud574 \ubcf4\uc790. \uc774 \ud568\uc218\ub294 \\(x=0\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\uace0 \\(x=2\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4. \\(x=-1\\)\uc5d0\uc11c\ub294 \ucd5c\ub313\uac12\uc774\ub098 \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \ub9cc\uc57d \uc5f4\ub9b0 \uad6c\uac04 \\(I = (-2,\\,0)\\)\uc744 \uc0dd\uac01\ud55c\ub2e4\uba74 \\(f\\)\ub294 \\(D_3 \\cap I\\) \uc704\uc5d0\uc11c \ucd5c\ub313\uac12 \\(f(-1)\\)\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \ud568\uc218\uc758 \uad6d\uc18c\uadf9\uac12\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"definition\">\n<p><span class=\"definition\">\uc815\uc758 2. (\ud568\uc218\uc758 \uadf9\ub313\uac12\uacfc \uadf9\uc19f\uac12)<\/span><\/p>\n<p>\\(f\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(D\\)\uc778 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(c_1 \\in D\\)\uc774\uace0, \\(c_1\\)\uc744 \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0\uad6c\uac04 \\(I_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(f\\)\uac00 \\(D \\cap I_1\\) \uc704\uc5d0\uc11c \ucd5c\ub313\uac12 \\(f(c_1 )\\)\uc744 \uac00\uc9c0\uba74 \u2018\\(f\\)\ub294 \\(c_1\\)\uc5d0\uc11c <span class=\"defined\">\uadf9\ub313\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(c_2 \\in D\\)\uc774\uace0, \\(c_2\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0\uad6c\uac04 \\(I_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f\\)\uac00 \\(D \\cap I_2\\) \uc704\uc5d0\uc11c \ucd5c\uc19f\uac12 \\(f(c_2 )\\)\ub97c \uac00\uc9c0\uba74 \u2018\\(f\\)\ub294 \\(c_2\\)\uc5d0\uc11c <span class=\"defined\">\uadf9\uc19f\uac12<\/span>\uc744 \uac00\uc9c4\ub2e4\u2019\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uadf9\ub313\uac12\uacfc \uadf9\uc19f\uac12\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uad6d\uc18c\uadf9\uac12<\/span>(local extrema) \ub610\ub294 <span class=\"defined\">\uc0c1\ub300\uadf9\uac12<\/span>(relative extrema)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uc804\uc5ed\uadf9\uac12<\/span> \ub610\ub294 <span class=\"defined\">\uc808\ub300\uadf9\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ucc45\uc5d0 \ub530\ub77c\uc11c <span class=\"defined\">\uadf9\uac12<\/span>\uc740 \uc0c1\ub300\uadf9\uac12\uc744 \uc774\ub974\uae30\ub3c4 \ud558\uace0 \uc808\ub300\uadf9\uac12\uc744 \uc774\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<p>\uc815\uc758\uc5d0 \uc758\ud558\uba74 \ubaa8\ub4e0 \uc808\ub300\uadf9\uac12\uc740 \uad6d\uc18c\uadf9\uac12\uc774\ub2e4. \uadf8\ub7ec\ub098 \uadf8 \uc5ed\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\ud398\ub974\ub9c8\uc758 \uc815\ub9ac; \uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc774 \\(D\\)\uc774\uace0 \\(c\\)\uac00 \\(D\\)\uc758 \ub0b4\uc810\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \uadf8 \uc810\uc5d0\uc11c \uad6d\uc18c\uadf9\uac12\uc744 \uac00\uc9c0\uba74 \\(f &#8216; (c)=0\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac16\ub294 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ub2e4\uc74c \uadf9\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4.<br \/>\n\\[f &#8216; (c) = \\lim_{x\\to c}\\frac{f(x)-f(c)}{x-c}.\\tag{1}\\]<br \/>\n\ud2b9\ud788 \\(c\\)\uac00 \\(D\\)\uc758 \ub0b4\uc810\uc774\ubbc0\ub85c \uadf9\ud55c (1)\uc740 \uc88c\uadf9\ud55c\uc774\ub098 \uc6b0\uadf9\ud55c\uc73c\ub85c \ubc14\uafb8\uc5b4\ub3c4 \ub3d9\uc77c\ud558\ub2e4. \uc989<br \/>\n\\[f &#8216; (c) = \\lim_{x\\to c^+}\\frac{f(x)-f(c)}{x-c} = \\lim_{x\\to c^-}\\frac{f(x)-f(c)}{x-c}.\\]<br \/>\n\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c0\ubbc0\ub85c \\(c\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04 \\(I = (a,\\,b)\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[x\\in I\\cap D \\quad \\Rightarrow \\quad f(x) \\le f(c)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(x \\in I\\cap D,\\) \\(x < c\\)\uc778 \uacbd\uc6b0\n\\[\\frac{f(x)-f(c)}{x-c} \\ge 0\\]\n\uc774\ubbc0\ub85c\n\\[f ' (c) = \\lim_{x\\to c^{-}}\\frac{f(x)-f(c)}{x-c} \\ge 0\\]\n\uc774\uba70, \\(x \\in I\\cap D,\\) \\(x > c\\)\uc778 \uacbd\uc6b0<br \/>\n\\[\\frac{f(x)-f(c)}{x-c} \\le 0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[f &#8216; (c) = \\lim_{x\\to c^{+}}\\frac{f(x)-f(c)}{x-c} \\le 0\\]<br \/>\n\uc774\ub2e4. \uc989 \\(0 \\le f &#8216; (c) \\le 0\\)\uc774\ubbc0\ub85c \\(f &#8216; (c)=0\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810 \\(x\\) \uc911\uc5d0\uc11c, \uadf8 \uc810\uc5d0\uc11c \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uac70\ub098, \\(f &#8216; (x) =0\\)\uc778 \uc810 \\(x\\)\ub97c \\(f\\)\uc758 <span class=\"defined\">\uc784\uacc4\uc810<\/span>(critical point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ub2eb\ud78c \uad6c\uac04 \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uae30 \uc704\ud574\uc11c\ub294 \uadf8 \uad6c\uac04\uc758 \ub05d\uc810\uacfc \\(f\\)\uc758 \ubaa8\ub4e0 \uc784\uacc4\uc810\uc5d0\uc11c \\(f\\)\uc758 \ud568\uc22b\uac12\uc744 \uad6c\ud55c \ub4a4, \uadf8 \uac12\ub4e4 \uc911 \uac00\uc7a5 \ud070 \uac12\uacfc \uac00\uc7a5 \uc791\uc740 \uac12\uc744 \ucc3e\uc73c\uba74 \ub41c\ub2e4.<\/p>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\ud3c9\uade0\uac12 \uc815\ub9ac<\/h3>\n<p>\uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\ub864\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f(a) = f(b)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f &#8216; (c) =0\\)\uc778 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p> \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc790\uba85\ud558\uac8c \\(f &#8216; (c) = 0\\)\uc778 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uac00 \uc544\ub2cc \uacbd\uc6b0\ub9cc \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(f\\)\ub294 \\([a,\\,b]\\) \uc704\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uc774\ub098 \ucd5c\uc19f\uac12 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(f(a)\\)\uc640 \ub2e4\ub978 \uac12\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc774 \ubaa8\ub450 \\(f(a)\\)\uc640 \uac19\ub2e4\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uac00 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\ub313\uac12\uc774 \\(f(a)\\)\uc640 \ub2e4\ub974\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(f(c)\\)\uac00 \\(f\\)\uc758 \ucd5c\ub313\uac12\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(c \\ne a,\\) \\(c\\ne b\\)\uc774\ubbc0\ub85c \\(c\\)\ub294 \\([a,\\,b]\\)\uc758 \ub0b4\uc810\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(f &#8216; (c) =0\\)\uc774\ub2e4.<\/p>\n<p>\\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\uc19f\uac12 \\(f(c)\\)\uac00 \\(f(a)\\)\uc640 \ub2e4\ub978 \uacbd\uc6b0\uc5d0\ub3c4 \ub3d9\uc77c\ud55c \ubc29\ubc95\uc73c\ub85c \\(f &#8216; (c)=0\\)\uc774\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub864\uc758 \uc815\ub9ac\ub97c \uc77c\ubc18\ud654\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 3. (\ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\frac{f(b)-f(a)}{b-a} = f &#8216; (c)\\tag{2}\\]<br \/>\n\uc778 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \ud568\uc218 \\(h\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[h(x) = f(x)-f(a) &#8211; \\frac{f(b)-f(a)}{b-a}(x-a).\\]<br \/>\n\uadf8\ub7ec\uba74 \\(h\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(h(a)=h(b)=0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub864\uc758 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\((a,\\,b)\\)\uc5d0 \uc810 \\(c\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(h &#8216; (c) = 0\\) \uc989<br \/>\n\\[h &#8216; (c) = f &#8216; (c) &#8211; \\frac{f(b)-f(a)}{b-a} = 0\\tag{3}\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. (3)\uc73c\ub85c\ubd80\ud130 (2)\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \ubbf8\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 1. (\uc0c1\uc218\ud568\uc218\uc758 \ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; (x)=0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(f\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\((a,\\,b)\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(x_1 ,\\) \\(x_2\\)\uc744 \uc0dd\uac01\ud558\uc790. \\(x_1 < x_2\\)\ub77c\uace0 \ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. (\ub450 \uc218\uc758 \ub300\uc18c \uad00\uacc4\uac00 \ubc18\ub300\ub77c\uba74, \ub450 \uc218\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.) \\(f\\)\ub294 \\([x_1 ,\\,x_2 ]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((x_1 ,\\,x_2 )\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(c\\in (x_1 ,\\,x_2 )\\)\uac00 \uc874\uc7ac\ud558\uc5ec\n\\[\\frac{f(x_2 ) - f(x_1 )}{x_2 - x_1} = f ' (c) =0\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f(x_2 ) = f(x_1 )\\)\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x_1\\)\uacfc \\(x_2\\)\ub294 \\((a,\\,b)\\)\uc5d0 \uc18d\ud558\ub294 \uc11c\ub85c \ub2e4\ub978 \uc784\uc758\uc758 \ub450 \uc810\uc774\ubbc0\ub85c \\(f\\)\ub294 \\((a,\\,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc77c\uc815\ud55c \ud568\uc22b\uac12\uc744 \uac00\uc9c4\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \ub4a4\uc5d0\uc11c \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud560 \ub54c, \uadf8\ub9ac\uace0 \uc801\ubd84\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 2.<\/span><\/p>\n<p>\ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \uc774 \uad6c\uac04\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f &#8216; (x) = g &#8216; (x)\\)\uc774\uba74 \\(f\\)\uc640 \\(g\\)\ub294 \uc774 \uad6c\uac04\uc5d0\uc11c \uc0c1\uc218 \ucc28\uc774\uc774\ub2e4. \uc989 \uc0c1\uc218 \\(C\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x \\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) = g(x)+C\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(h = f-g\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\((a,\\,b)\\)\uc5d0\uc11c \\(h &#8216; = f &#8216; &#8211; g &#8216; = 0\\)\uc774\ubbc0\ub85c \ub530\ub984\uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \\(h\\)\ub294 \\((a,\\,b)\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc774\ub2e4. \uc989 \uc0c1\uc218 \\(C\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(h(x) = C\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) = g(x)+C\\)\uc774\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(x_1 < x_2\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ub450 \uc810 \\(x_1 ,\\) \\(x_2\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_1 ) \\le f(x_2 )\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uc99d\uac00<\/span>\ud55c\ub2e4\u2019\uace0 \ub9d0\ud558\uace0, \ub9cc\uc57d \\(x_1 < x_2\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ub450 \uc810 \\(x_1 ,\\) \\(x_2\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_1 ) \\ge f(x_2 )\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uac10\uc18c<\/span>\ud55c\ub2e4\u2019\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(x_1 < x_2\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ub450 \uc810 \\(x_1 ,\\) \\(x_2\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_1 ) < f(x_2 )\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uc21c\uc99d\uac00<\/span>\ud55c\ub2e4\u2019\uace0 \ub9d0\ud558\uace0, \ub9cc\uc57d \\(x_1 < x_2\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ub450 \uc810 \\(x_1 ,\\) \\(x_2\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_1 ) > f(x_2 )\\)\uac00 \uc131\ub9bd\ud558\uba74 \u2018\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uc21c\uac10\uc18c<\/span>\ud55c\ub2e4\u2019\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ub3c4\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uc99d\uac10\uc744 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 3. (\ud568\uc218\uc758 \uc99d\uac10\uc5d0 \ub300\ud55c \ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f &#8216; (x) \\ge 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc99d\uac00\ud55c\ub2e4. \ud2b9\ud788 \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f &#8216; (x) > 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud55c\ub2e4.<\/li>\n<li>\ub9cc\uc57d \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f &#8216; (x) \\le 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uac10\uc18c\ud55c\ub2e4. \ud2b9\ud788 \uc784\uc758\uc758 \\(x\\in (a,\\,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f &#8216; (x) < 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc21c\uac10\uc18c\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(x_1\\)\uacfc \\(x_2\\)\uac00 \\([a,\\,b]\\)\uc758 \uc810\uc774\uace0 \\(x_1 < x_2\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\)\ub294 \\([x_1 ,\\,x_2]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((x_1 ,\\,x_2 )\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec\n\\[\\frac{f(x_2 ) - f(x_1 )}{x_2 - x_1} = f ' (c)\\tag{4}\\]\n\uc778 \uc810 \\(c\\)\uac00 \\((x_1 ,\\,x_2 )\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. (4)\uc758 \uc88c\ubcc0\uc5d0\uc11c \ubd84\ubaa8\ub294 \\(x_2 - x_1 > 0\\)\uc774\ubbc0\ub85c, \\(f(x_2 ) &#8211; f(x_1 )\\)\uc758 \ubd80\ud638\ub294 \\(f &#8216; (c)\\)\uc758 \ubd80\ud638\uc640 \uc77c\uce58\ud55c\ub2e4. \uc989 \\(f &#8216; (c) > 0\\)\uc774\uba74 \\(f(x_2 ) > f(x_1 )\\)\uc774\uba70, \\(f &#8216; (c) < 0\\)\uc774\uba74 \\(f(x_2 ) < f(x_1 )\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(x_1 ,\\) \\(x_2\\)\ub294 \\([a,\\,b]\\)\uc758 \uc784\uc758\uc758 \ub450 \uc810\uc774\ubbc0\ub85c, \\((a,\\,b)\\)\uc5d0\uc11c \\(f ' > 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc99d\uac00\ud558\uace0, \\((a,\\,b)\\)\uc5d0\uc11c \\(f &#8216; < 0\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uac10\uc18c\ud55c\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p><!-- ##### ##### ##### ##### ##### ##### ##### ##### ##### ##### --><\/p>\n<h3>\ub3c4\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc131\uc9c8<\/h3>\n<p>\ud568\uc218 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\ub294 \uc5f0\uc18d\uc774\uc9c0\ub9cc, \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud574\uc11c \\(f &#8216; \\)\uc774 \uc5f0\uc18d\uc774\ub77c\ub294 \ubcf4\uc7a5\uc740 \uc5c6\ub2e4. \uadf8\ub7ec\ub098 \ub180\ub78d\uac8c\ub3c4 \ubaa8\ub4e0 \ub3c4\ud568\uc218\ub294 <span class=\"defined\">\uc0ac\uc787\uac12 \uc131\uc9c8<\/span>(intermediate property)\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 4. (\ub2e4\ub974\ubd80\uc758 \uc815\ub9ac; \ub3c4\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\(I = [a,\\,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc0ac\uc787\uac12 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4. \uc989 \\(f &#8216; (a) \\ne f &#8216; (b)\\)\uc774\uace0 \\(C\\)\uac00 \\(f &#8216; (a)\\)\uc640 \\(f &#8216; (b)\\) \uc0ac\uc774\uc5d0 \uc788\ub294 \uac12\uc774\uba74 \\(f &#8216; (x_0) = C\\)\uc778 \uc810 \\(x_0\\)\uc774 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f &#8216; (a) > f &#8216; (b)\\)\uc778 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud574\ub3c4 \ucda9\ubd84\ud558\ub2e4. \uc989 \\(f &#8216; (a) > C > f &#8216; (b)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(I\\)\uc5d0\uc11c \ud568\uc218 \\(h\\)\ub97c<br \/>\n\\[h(x) = f(x) &#8211; Cx\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \\(h\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uad6c\uac04\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4. \\(h &#8216; (a) = f &#8216; (a) &#8211; C > 0\\)\uc774\ubbc0\ub85c \\(h\\)\ub294 \\(a\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(h\\)\uac00 \\(a\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\uba74 \uc784\uc758\uc758 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\frac{h(x) &#8211; h(a)}{x-a} \\le 0\\]<br \/>\n\uc774\ubbc0\ub85c \\(h &#8216; (a) \\le 0\\)\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(h\\)\ub294 \\(b\\)\uc5d0\uc11c\ub3c4 \ucd5c\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(h\\)\ub294 \\([a,\\,b]\\)\uc758 \ub0b4\uc810, \uc989 \\((a,\\,b)\\)\uc5d0 \uc18d\ud558\ub294 \uc810\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uac00\uc9c4\ub2e4. \\(h\\)\uac00 \ucd5c\ub313\uac12\uc744 \uac16\ub294 \uc810\uc744 \\(x_0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc0c1\ub300\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud558\uc5ec \\(h &#8216; (x_0) =0\\) \uc989 \\(f &#8216; (x_0 ) = C\\)\uac00 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc9c0\uae08\uae4c\uc9c0 \ud568\uc218 \\(f\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \ubbf8\ubd84\uc758 \uc815\uc758\uc640 \ubbf8\ubd84\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \\(f\\)\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud558\uace0 \\(f\\) \ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uc5ec\ub7ec \uac00\uc9c0 \ubc29\ubc95\uc744 \uc0b4\ud3b4 \ubcf4\uc558\ub2e4. \uadf8\ub807\ub2e4\uba74 \uc5ed\uc73c\ub85c \uc5b4\ub5a0\ud55c \ud568\uc218\uac00 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c \uadf8 \ud568\uc218\uac00 \ub2e4\ub978 \ud568\uc218\uc758 \ub3c4\ud568\uc218\uac00 \ub420 \uc218 \uc788\ub294\uc9c0 \uc5ec\ubd80\ub294 \uc5b4\ub5bb\uac8c \ud310\ub2e8\ud560 \uc218 \uc788\uc744\uae4c? \ub2e4\uc74c \uc815\ub9ac\ub294 \uadf8\uc5d0 \ub300\ud55c \ubd80\ubd84\uc801\uc778 \ub2f5\uc744 \uc900\ub2e4.<\/p>\n<div class=\"corollary\">\n<p><span class=\"corollary\">\ub530\ub984\uc815\ub9ac 4.<\/span><\/p>\n<p>\ub3c4\ud568\uc218\ub294 \ub2e8\uc21c\ubd88\uc5f0\uc18d\uc810\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216;\\)\uc774 \\(I\\)\uc758 \ub0b4\uc810 \\(c\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f &#8216; \\)\uc774 \\(c\\)\uc5d0\uc11c \ub2e8\uc21c\ubd88\uc5f0\uc18d\uc774\ub77c\uba74 \\(c\\)\uc5d0\uc11c \\(f &#8216; \\)\uc758 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc774 \uc874\uc7ac\ud558\uc9c0\ub9cc \uadf8 \uac12\uc740 \ub2e4\ub974\ub2e4. \\(c\\)\uc5d0\uc11c \\(f &#8216; \\)\uc758 \uc88c\uadf9\ud55c\uc744 \\(L,\\) \uc6b0\uadf9\ud55c\uc744 \\(R\\)\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(L < R\\)\uc778 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. (\ub9cc\uc57d \ubd80\ub4f1\ud638\uac00 \ubc18\ub300\ub85c \uc131\ub9bd\ud55c\ub2e4\uba74 \\(L\\)\uacfc \\(R\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \ub41c\ub2e4.) \\(\\epsilon = \\frac{1}{3}(R-L)\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uadf9\ud55c\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(\\delta_L > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x \\in (c-\\delta_L ,\\, c)\\)\uc77c \ub54c\ub9c8\ub2e4 \\(f &#8216; (x) < L+\\epsilon\\)\uc774 \uc131\ub9bd\ud558\uba70, \\(\\delta_R > 0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x \\in (c ,\\,c+\\delta_R )\\)\uc77c \ub54c\ub9c8\ub2e4 \\(f &#8216; (x) > R-\\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\\([a,\\,b] = [c-\\frac{1}{2}\\delta_L ,\\, c+\\frac{1}{2}\\delta_R ]\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec<\/p>\n<p class=\"aligncenter\">\\(f &#8216; (x) < L+\\epsilon\\) \ub610\ub294 \\(f ' (x) > R-\\epsilon\\) \ub610\ub294 \\(f &#8216; (x) = f &#8216; (c)\\)<\/p>\n<p>\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(L+\\epsilon < R-\\epsilon\\)\uc774\ubbc0\ub85c, \\(C \\in (L+\\epsilon ,\\, R-\\epsilon ),\\) \\(C \\ne f ' (c)\\)\uc778 \uac12 \\(C\\)\ub97c \ud0dd\ud558\uba74 \\(f ' (a) < C < f ' (b)\\)\uc774\uc9c0\ub9cc \uc784\uc758\uc758 \\(x_0 \\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f ' (x_0) \\ne C\\)\uc774\ub2e4. \uc989 \\(f ' \\)\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \uc0ac\uc787\uac12 \uc131\uc9c8\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc774\uac83\uc740 \ub2e4\ub974\ubd80\uc758 \uc815\ub9ac\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f ' \\)\uc740 \\(c\\)\uc5d0\uc11c \ub2e8\uc21c\ubd88\uc5f0\uc18d\uc77c \uc218 \uc5c6\ub2e4.\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<h3>\uc7a1\uc124<\/h3>\n<p>\uae00\uc744 \uc4f0\uace0 \ubcf4\ub2c8 \ubcf4\uae30\uc640 \uc608\uc81c\uac00 \uc5c6\ub124\uc694. \ub2e4\uc74c\uc5d0 \ub123\uaca0\uc2b5\ub2c8\ub2e4.<\/p>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\uc608\uc81c n.<\/span>\n...\n<\/p>\n\n\n\n\n\n<p><span class=\"proof\">\ud480\uc774.<\/span>\n...\n<span class=\"qee\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac n.<\/span><\/p>\n\n\n\n\n<p>..<\/p>\n\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85<\/p>\n\n\n\n\n<p>...\n<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \ub0b4\uc810 \\(a\\)\uc5d0\uc11c\uc758 \ubbf8\ubd84\uacc4\uc218 \\(f &#8216; (a)\\)\ub294 \\(x=a\\)\uc77c \ub54c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \uae30\uc6b8\uae30\uc640 \uac19\ub2e4. \uc989 \ubbf8\ubd84\uacc4\uc218\ub294 \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \ubaa8\uc591\uc5d0 \uc758\ud558\uc5ec \uacb0\uc815\ub418\ubbc0\ub85c, \ubbf8\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \ud568\uc218\uc758 \uadf8\ub798\ud504\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \uc218 \uc788\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud568\uc218\uc758 \uadf9\uac12, \uc99d\uac00\uc640 \uac10\uc18c\ub97c \uc815\uc758\ud558\uace0 \uc774\uc640 \uad00\ub828\ub41c \ubbf8\ubd84\uc758 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ud568\uc218\uc758 \uadf9\uac12 \\(E\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(M \\in E\\)\uc774\uace0, \uc784\uc758\uc758 \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\le M\\)\uc774\uba74 \\(M\\)\uc744 \\(E\\)\uc758 \ucd5c\ub313\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.&hellip;<\/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":[244,254,251,247,243,242,239,245,249,240,252,250,248,253,241,246,238],"class_list":["post-1962","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-absolute-extrema","tag-darboux-theorem","tag-derivative-test","tag-fermats-theorem","tag-global-extrema","tag-local-extrema","tag-mean-value-theorem","tag-relative-extrema","tag-rolles-theorem","tag-240","tag-252","tag-250","tag-248","tag-253","tag-241","tag-246","tag-238"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1962","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=1962"}],"version-history":[{"count":35,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1962\/revisions"}],"predecessor-version":[{"id":3126,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1962\/revisions\/3126"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1962"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1962"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1962"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}