{"id":9486,"date":"2025-10-20T18:55:25","date_gmt":"2025-10-20T09:55:25","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9486"},"modified":"2026-09-27T14:45:14","modified_gmt":"2026-09-27T05:45:14","slug":"ch05-differentiation-of-functions-of-one-variable","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\/","title":{"rendered":"\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n\n --><\/p>\n<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uc2e4\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \uc815\uc758\ud558\uace0, \ud3c9\uade0\uac12 \uc815\ub9ac, \ud14c\uc77c\ub7ec \uc815\ub9ac, \ubcfc\ub85d\uc131 \ub4f1 \uc77c\ubcc0\uc218 \ubbf8\ubd84\ubc95\uc758 \uc8fc\uc694 \uacb0\uacfc\ub97c \ub2e4\ub8ec\ub2e4.<\/p>\n<h3>\ubbf8\ubd84 \uac00\ub2a5\uc131<\/h3>\n<p>\\(X\\)\uac00 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(c\\in X\\cap X&#8217;\\)\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f\\colon X\\to\\mathbb R\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c <span class=\"defined\">\ubbf8\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(differentiable)\ub294 \uac83\uc740 \\(X\\) \uc548\uc5d0\uc11c \ub2e4\uc74c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.<br \/>\n\\[<br \/>\nf'(c)=\\lim_{x\\to c}\\frac{f(x)-f(c)}{x-c}.<br \/>\n\\]<br \/>\n\uc774 \uadf9\ud55c\uac12 \\(f'(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\ubbf8\ubd84\uacc4\uc218<\/span>(derivative)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\\(h=x-c\\)\ub85c \ub193\uc73c\uba74, \\(c+h\\in X\\)\uc778 \\(h\\)\uc5d0 \ub300\ud558\uc5ec \uc704 \uc815\uc758\ub97c<br \/>\n\\[<br \/>\nf'(c)=\\lim_{h\\to0}\\frac{f(c+h)-f(c)}{h}<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \ub530\ub77c\uc11c \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uac83\uc740 \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(L\\)\uacfc \ud568\uc218 \\(r\\)\uac00 \uc874\uc7ac\ud558\ub294 \uac83\uacfc \ub3d9\uce58\uc774\ub2e4.<br \/>\n\\[<br \/>\nf(c+h)=f(c)+Lh+r(h),\\quad<br \/>\n\\lim_{\\substack{h\\to0\\\\c+h\\in X}}\\frac{r(h)}{h}=0.<br \/>\n\\]<br \/>\n\uc774\ub54c \\(L=f'(c)\\)\uc774\ub2e4.<\/p>\n<p>\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc9c1\uc120 \\(y=f'(c)(x-c)+f(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc811\uc120<\/span>(tangent line)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f\\colon X\\to\\mathbb R\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810 \uc911 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc810\uc758 \uc9d1\ud569\uc744 \\(D\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f&#8217;\\)\uc740 \\(D\\)\uc758 \uc810 \\(x\\)\ub97c \\(f'(x)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(f&#8217;\\)\uc744 \\(f\\)\uc758 <span class=\"defined\">\ub3c4\ud568\uc218<\/span>(derivative function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.1.<\/span><br \/>\n\ubbf8\ubd84\uacc4\uc218\uc640 \ub3c4\ud568\uc218\uc758 \ucc28\uc774\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.1. (\ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uc5f0\uc18d\uc131\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \\(f\\)\ub294 \uc810 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(x\\neq c\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nf(x)-f(c)=\\frac{f(x)-f(c)}{x-c}(x-c)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \\(x\\to c\\)\uc77c \ub54c \uccab \ubc88\uc9f8 \uc778\uc790\ub294 \\(f'(c)\\)\ub85c \uc218\ub834\ud558\uace0 \ub450 \ubc88\uc9f8 \uc778\uc790\ub294 \\(0\\)\uc73c\ub85c \uc218\ub834\ud558\ubbc0\ub85c \\(f(x)-f(c)\\to0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f(x)\\to f(c)\\), \uc989 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uadf8\ub7ec\ub098 \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \ubaa8\ub450 \ubbf8\ubd84 \uac00\ub2a5\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(f(x)=|x|\\)\ub294 \\(x=0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\uc2e4\ud568\uc218\uc758 \uadf9\ud55c\uc5d0\uc11c \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc744 \uc815\uc758\ud55c \uac83\ucc98\ub7fc, \ubbf8\ubd84\uacc4\uc218\ub3c4 <span class=\"defined\">\ud55c\ubc29\ud5a5 \ubbf8\ubd84\uacc4\uc218<\/span>\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc88c\ubbf8\ubd84\uacc4\uc218<\/span>: \\(\\displaystyle f&#8217;_-(c)=\\lim_{h\\to0^-}\\frac{f(c+h)-f(c)}{h}\\)<\/li>\n<li><span class=\"defined\">\uc6b0\ubbf8\ubd84\uacc4\uc218<\/span>: \\(\\displaystyle f&#8217;_+(c)=\\lim_{h\\to0^+}\\frac{f(c+h)-f(c)}{h}\\)<\/li>\n<\/ul>\n<p>\ub2eb\ud78c\uad6c\uac04\uc758 \ub05d\uc810\uc5d0\uc11c\ub294 \ud55c\ubc29\ud5a5 \ubbf8\ubd84\uacc4\uc218\ub85c \ubbf8\ubd84\uc744 \uc815\uc758\ud55c\ub2e4. \uc989 \ud568\uc218 \\(f\\colon[a,b]\\to\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec, \\(f'(a)\\)\uc640 \\(f'(b)\\)\ub294 \uac01\uac01 \uc6b0\ubbf8\ubd84\uacc4\uc218\uc640 \uc88c\ubbf8\ubd84\uacc4\uc218\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n<p>\uad6c\uac04 \ub0b4\ubd80\uc758 \uc810\uc5d0\uc11c \ud568\uc218\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc88c\ubbf8\ubd84\uacc4\uc218\uc640 \uc6b0\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \uc874\uc7ac\ud558\uace0 \ub450 \ubbf8\ubd84\uacc4\uc218\uac00 \uc77c\uce58\ud558\ub294 \uac83\uc774\ub2e4. [\uc88c\ubbf8\ubd84\uacc4\uc218\uc640 \ub3c4\ud568\uc218\uc758 \uc88c\uadf9\ud55c\uc740 \uc11c\ub85c \ub2e4\ub97c \uc218 \uc788\ub2e4.]<\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(f&#8217;\\)\uc740 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4<br \/>\n\\[<br \/>\nf(x)=<br \/>\n\\begin{cases}<br \/>\nx^2\\sin\\dfrac1x &#038; \\text{if }\\;x\\neq0,\\\\[5pt]<br \/>\n0 &#038; \\text{if }\\;x=0<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(f&#8217;\\)\uc740 \\(x=0\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 5.2. (\ubbf8\ubd84\uc758 \uc5f0\uc0b0\ubc95\uce59)<\/span><\/p>\n<p>\\(f\\)\uc640 \\(g\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(\\alpha,\\beta\\in\\mathbb R\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\((\\alpha f+\\beta g)'(c)=\\alpha f'(c)+\\beta g'(c)\\).<\/li>\n<li>\\((fg)'(c)=f'(c)g(c)+f(c)g'(c)\\).<\/li>\n<li>\\(g(c)\\neq0\\)\uc774\uba74 \\(\\displaystyle(f\/g)'(c)=\\frac{f'(c)g(c)-f(c)g'(c)}{g(c)^2}\\).<\/li>\n<li>\ud569\uc131 \\(g\\circ f\\)\uac00 \\(c\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0, \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(g\\)\uac00 \\(f(c)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74<br \/>\n\\[<br \/>\n(g\\circ f)'(c)=g'(f(c))f'(c).<br \/>\n\\]<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc120\ud615\uc131\uc740 \ucc28\ubd84\ubaab\uc5d0 \uadf9\ud55c\uc758 \uc120\ud615\uc131\uc744 \uc801\uc6a9\ud558\uba74 \ubc14\ub85c \uc5bb\ub294\ub2e4. \uacf1\uc758 \ubc95\uce59\uc740<br \/>\n\\[<br \/>\n\\frac{f(x)g(x)-f(c)g(c)}{x-c}<br \/>\n=<br \/>\nf(x)\\frac{g(x)-g(c)}{x-c}<br \/>\n+<br \/>\ng(c)\\frac{f(x)-f(c)}{x-c}<br \/>\n\\]<br \/>\n\uc5d0 \\(x\\to c\\)\ub97c \ucde8\ud558\uba74 \uc5bb\ub294\ub2e4. \ubbf8\ubd84 \uac00\ub2a5\uc131\uc740 \uc5f0\uc18d\uc131\uc744 \ud568\uc758\ud558\ubbc0\ub85c \\(f(x)\\to f(c)\\)\uc774\ub2e4.<\/p>\n<p>\\(g(c)\\neq0\\)\uc774\uba74 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(c\\)\uc758 \ucda9\ubd84\ud788 \uc791\uc740 \uadfc\ubc29\uc5d0\uc11c \\(g(x)\\neq0\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\frac{1\/g(x)-1\/g(c)}{x-c}<br \/>\n=<br \/>\n-\\frac{1}{g(x)g(c)}<br \/>\n\\frac{g(x)-g(c)}{x-c}<br \/>\n\\]<br \/>\n\uc774\uace0, \uc5ec\uae30\uc11c \uc5ed\uc218\uc758 \ubbf8\ubd84\ubc95\uacfc \uacf1\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uba74 \ubaab\uc758 \ubc95\uce59\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9\uc73c\ub85c \\(L=g'(f(c))\\)\ub77c\uace0 \ud558\uace0<br \/>\n\\[<br \/>\n\\eta(y)=<br \/>\n\\begin{cases}<br \/>\n\\dfrac{g(y)-g(f(c))}{y-f(c)}-L,&#038; y\\neq f(c),\\\\[6pt]<br \/>\n0,&#038; y=f(c)<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\ub77c\uace0 \ub193\uc73c\uba74 \\(y\\to f(c)\\)\uc77c \ub54c \\(\\eta(y)\\to0\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\ng(f(x))-g(f(c))<br \/>\n=<br \/>\n\\bigl(L+\\eta(f(x))\\bigr)\\bigl(f(x)-f(c)\\bigr).<br \/>\n\\]<br \/>\n\uc774\ub97c \\(x-c\\)\ub85c \ub098\ub204\uace0 \\(x\\to c\\)\ub97c \ucde8\ud558\uba74 \uc5f0\uc1c4\ubc95\uce59\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud2b9\ud788 \ub2e4\ud56d\ud568\uc218 \\(f(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\)\uc758 \ub3c4\ud568\uc218\ub294<br \/>\n\\[<br \/>\nf'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\\cdots+a_1<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.2.<\/span><br \/>\n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc77c \ub54c, \uacf1\uc758 \ubc95\uce59\uc744 \uc0ac\uc6a9\ud558\uc5ec<br \/>\n\\[<br \/>\n(f^n)&#8217;=nf^{\\,n-1}f&#8217;<br \/>\n\\]<br \/>\n\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.3.<\/span><br \/>\n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(f(x)=x^n\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nf'(x)=nx^{n-1}<br \/>\n\\]<br \/>\n\uc784\uc744 \ubbf8\ubd84\uc758 \uc815\uc758\uc5d0\uc11c \uc9c1\uc811 \uc99d\uba85\ud558\uc2dc\uc624. \uc5ec\uae30\uc11c \\(x^0=1\\)\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.4.<\/span><br \/>\n\\(m\\)\uc774 \\(0\\)\uc774 \uc544\ub2cc \uc815\uc218\ub77c\uace0 \ud558\uc790. \\(f(x)=x^m\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nf'(x)=mx^{m-1}<br \/>\n\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. \ub2e8, \\(m&lt;0\\)\uc774\uba74 \\(x\\neq0\\)\uc778 \uc810\uc5d0\uc11c \uc0dd\uac01\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.5.<\/span><br \/>\n\ubbf8\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624. (<a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ubb38\uc81c 4.8<\/a>\uc744 \ucc38\uc870\ud558\uc2dc\uc624.)<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(x)=\\sin x\\)<\/li>\n<li>\\(f(x)=\\cos x\\)<\/li>\n<li>\\(f(x)=\\tan x\\) (\ub2e8, \\(x\\neq\\frac{2k+1}{2}\\pi\\), \\(k\\in\\mathbb Z\\))<\/li>\n<li>\\(f(x)=\\sec x\\) (\ub2e8, \\(x\\neq\\frac{2k+1}{2}\\pi\\), \\(k\\in\\mathbb Z\\))<\/li>\n<li>\\(f(x)=\\csc x\\) (\ub2e8, \\(x\\neq k\\pi\\), \\(k\\in\\mathbb Z\\))<\/li>\n<li>\\(f(x)=\\cot x\\) (\ub2e8, \\(x\\neq k\\pi\\), \\(k\\in\\mathbb Z\\))<\/li>\n<\/ol>\n<\/div>\n<h3>\ud3c9\uade0\uac12 \uc815\ub9ac\uc640 \uadf8 \ub530\ub984\uc815\ub9ac<\/h3>\n<p>\ud3c9\uade0\uac12 \uc815\ub9ac\ub294 \ub3c4\ud568\uc218\uc758 \ud575\uc2ec\uc801\uc778 \uc131\uc9c8\uc744 \ub098\ud0c0\ub0b4\uba70, \ub3c4\ud568\uc218\ub97c \ud65c\uc6a9\ud558\ub294 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.3. (\uadf9\uac12\uc5d0 \ub300\ud55c \ud398\ub974\ub9c8\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(c\\)\uac00 \uad6c\uac04 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f'(c)=0\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\delta&gt;0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(|x-c|&lt;\\delta\\)\uc77c \ub54c \\(f(x)\\le f(c)\\)\uc774\ub2e4. \uc989 \\(|h|&lt;\\delta\\)\uc77c \ub54c \\(f(c+h)-f(c)\\le0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nf_-&#8216;(c)&#038;=\\lim_{h\\to0-}\\frac{f(c+h)-f(c)}{h}\\ge0,\\\\<br \/>\nf_+'(c)&#038;=\\lim_{h\\to0+}\\frac{f(c+h)-f(c)}{h}\\le0.<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uadf8\ub7f0\ub370 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\(f'(c)=f_-&#8216;(c)=f_+'(c)\\)\uc774\uace0, \ub530\ub77c\uc11c \\(f'(c)=0\\)\uc774\ub2e4. \uadf9\uc19f\uac12\uc758 \uacbd\uc6b0\uc5d0\ub294 \ubd80\ub4f1\ud638\uac00 \ubc18\ub300\uac00 \ub418\uba70 \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.4. (\ub864\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(a&lt;b\\)\uc774\uace0 \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 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(f'(c)=0\\)\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uc774\uba74 \\((a,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f&#8217;=0\\)\uc774\ubbc0\ub85c \uc790\uba85\ud558\ub2e4. \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uac00 \uc544\ub2c8\ub77c\uace0 \ud558\uc790. \ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac\uc5d0 \uc758\ud574 \\(f\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(f(a)=f(b)\\)\uc774\uace0 \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uac00 \uc544\ub2c8\ubbc0\ub85c \ucd5c\ub313\uac12\uc774\ub098 \ucd5c\uc19f\uac12 \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \ub0b4\ubd80\uc810 \\(c\\in(a,b)\\)\uc5d0\uc11c \uc5bb\uc5b4\uc9c4\ub2e4. \uc815\ub9ac 5.3\uc5d0 \uc758\ud574 \\(f'(c)=0\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.5. (\ub77c\uadf8\ub791\uc8fc \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\\(a&lt;b\\)\uc774\uace0 \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 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nf'(c)=\\frac{f(b)-f(a)}{b-a}<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uad6c\uac04 \\([a,b]\\) \uc704\uc5d0\uc11c \ud568\uc218 \\(g\\)\ub97c<br \/>\n\\[<br \/>\ng(x)=f(x)-\\frac{f(b)-f(a)}{b-a}(x-a)<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(g(a)=g(b)=f(a)\\)\uc774\ub2e4. \ub864\uc758 \uc815\ub9ac\ub97c \\(g\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud3c9\uade0\uac12 \uc815\ub9ac\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294 \uace1\uc120 \uc704\uc758 \uc5b4\ub5a4 \uc810\uc5d0\uc11c \uc811\uc120\uc758 \uae30\uc6b8\uae30\uac00 \ud560\uc120\uc758 \uae30\uc6b8\uae30\uc640 \uac19\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.6.<\/span><br \/>\n\ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ub4f1\uc2dd \\(|\\sin x-\\sin y|\\leq|x-y|\\)\uac00 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.6. (\ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n<p>\\(a&lt;b\\)\uc774\uace0 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \ubaa8\ub4e0 \\(x\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(g'(x)\\neq0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\frac{f'(c)}{g'(c)}=\\frac{f(b)-f(a)}{g(b)-g(a)}<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(g(b)\\neq g(a)\\)\uc784\uc744 \ud655\uc778\ud558\uc790. \ub9cc\uc57d \\(g(b)=g(a)\\)\ub77c\uba74 \ub864\uc758 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0\uc11c \\(g'(c)=0\\)\uc774 \ub418\ub294\ub370, \uc774\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \ud568\uc218 \\(h\\)\ub97c<br \/>\n\\[<br \/>\nh(x)=f(x)-\\frac{f(b)-f(a)}{g(b)-g(a)}g(x)<br \/>\n\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(h\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nh(a)&#038;=f(a)-\\frac{f(b)-f(a)}{g(b)-g(a)}g(a),\\\\<br \/>\nh(b)&#038;=f(b)-\\frac{f(b)-f(a)}{g(b)-g(a)}g(b)<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(h(a)=h(b)\\)\uc774\ub2e4. \\(h\\)\uc5d0 \ub864\uc758 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec \\(h'(c)=0\\)\uc774\ub2e4. \uc989<br \/>\n\\[<br \/>\nf'(c)-\\frac{f(b)-f(a)}{g(b)-g(a)}g'(c)=0.<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(g'(c)\\neq0\\)\uc774\ubbc0\ub85c \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud568\uc22b\uac12\uc758 \ubcc0\ud654\uc640 \uad00\ub828\ub41c \uc720\uc6a9\ud55c \uacf5\uc2dd\uc744 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 5.7. (\ud568\uc218\uc758 \uc99d\uac10\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)=0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc774\ub2e4.<\/li>\n<li>\\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)\\ge0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4.<\/li>\n<li>\\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)\\le0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4.<\/li>\n<li>\\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)&gt;0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud55c\ub2e4.<\/li>\n<li>\\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)&lt;0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uac10\uc18c\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(x&lt;y\\)\uc778 \\(I\\)\uc758 \ub450 \uc810\uc744 \ud0dd\ud558\uc790. \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(c\\in(x,y)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nf(y)-f(x)=f'(c)(y-x)<br \/>\n\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(f&#8217;\\)\uc758 \ubd80\ud638\uc5d0 \ub530\ub77c \\(f(y)-f(x)\\)\uc758 \ubd80\ud638\uac00 \uacb0\uc815\ub418\uba70 (1)\u2013(5)\uac00 \ubaa8\ub450 \ub530\ub978\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.7.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f\\)\uac00 \ub2e8\uc870\uc99d\uac00\ud558\uba74 \ubaa8\ub4e0 \\(x\\in I\\)\uc5d0\uc11c \\(f'(x)\\ge0\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/li>\n<li>\\(f\\)\uac00 \uc21c\uc99d\uac00\ud558\ub354\ub77c\ub3c4 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x)&gt;0\\)\uc77c \ud544\uc694\ub294 \uc5c6\uc74c\uc744 \ubc18\ub840\ub85c \ubcf4\uc774\uc2dc\uc624.<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.8.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc591\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(|f'(x)|\\leq M\\)\uc774\uba74, \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h3>\ud14c\uc77c\ub7ec \uc815\ub9ac<\/h3>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ub3c4\ud568\uc218 \\(f&#8217;\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \\(f&#8217;\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \\(f&#8221;(a)\\) \ub610\ub294 \\(\\frac{d^2}{dx^2}f(a)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc77c\ubc18\uc801\uc73c\ub85c \uc790\uc5f0\uc218 \\(n\\ge2\\)\uc5d0 \ub300\ud558\uc5ec \\(f^{(n-1)}\\)\uc774 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74<br \/>\n\\[<br \/>\nf^{(n)}(a)=(f^{(n-1)})'(a)<br \/>\n\\]<br \/>\n\ub97c \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uc810 \\(a\\)\uc5d0\uc11c \\(f^{(1)}(a),\\ldots,f^{(n)}(a)\\)\uac00 \ucc28\ub840\ub85c \uc815\uc758\ub420 \ub54c \u201c\\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\\(f^{(n)}(x)\\)\uac00 \uc815\uc758\ub418\ub294 \uc810 \\(x\\)\ub97c \\(f^{(n)}(x)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \\(f\\)\uc758 <span class=\"defined\">\\(n\\)\uacc4\ub3c4\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(f^{(n)}\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ud3b8\uc758\uc0c1 \\(f^{(0)}=f\\)\ub85c \ub454\ub2e4.<\/p>\n<p>\uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f&#8217;\\)\uc774 \uc5f0\uc18d\uc77c \ub54c, \u201c\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\\(C^1\\)<\/span>\uc774\ub2e4\u201d \ub610\ub294 \u201c\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uc5f0\uc18d\uc801\uc73c\ub85c \ubbf8\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(continuously differentiable)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. [\u201c\uc5f0\uc18d\uc801\uc73c\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\u201d\ub77c\ub294 \ub9d0\uc774 \u201c\uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\u201d\ub77c\ub294 \ub73b\uc774 \uc544\ub2c8\ub2e4. \uc774\ub7ec\ud55c \ud63c\ub3d9\uc744 \ud53c\ud558\ub824\uba74 \u201c\uc5f0\uc18d\uc778 \ub3c4\ud568\uc218\ub97c \uac00\uc9c4\ub2e4\u201d\ub77c\ub294 \ud45c\ud604\uc744 \uc0ac\uc6a9\ud558\ub294 \ud3b8\uc774 \ub0ab\ub2e4.] \ub610\ud55c \uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4\ub3c4\ud568\uc218\uac00 \uc874\uc7ac\ud558\uace0 \\(f^{(n)}\\)\uc774 \uc5f0\uc18d\uc77c \ub54c, \u201c\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\\(C^n\\)<\/span>\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \\(C^n\\)\uc774\uba74, \u201c\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \\(C^\\infty\\)\uc774\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c, \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec, \\(a\\) \uadfc\ucc98\uc5d0\uc11c \ub2e4\ud56d\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(f\\)\uc758 \uadfc\uc0ac\ud568\uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box definition\">\n<p><span class=\"definition\">\uc815\uc758 5.8. (\ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc5d0\uc11c \\(n\\)\ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\(a\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \\(n\\)\ucc28 <span class=\"defined\">\ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd<\/span>(Taylor polynomial)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\nP_n(x)=\\sum_{k=0}^{n}\\frac{f^{(k)}(a)}{k!}(x-a)^k.<br \/>\n\\]<\/p>\n<\/div>\n<p>\ub9ce\uc740 \uacbd\uc6b0\uc5d0 \\(P_n(x)\\)\uc758 \ucc28\uc218\uac00 \ud074\uc218\ub85d, \uadf8\ub9ac\uace0 \\(x\\)\uac00 \\(a\\)\uc5d0 \uac00\uae4c\uc6b8\uc218\ub85d, \\(P_n(x)\\)\uc758 \uac12\uc774 \\(f(x)\\)\uc758 \uac12\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4. \uc774\ub7ec\ud55c \uc0c1\ud669\uc5d0\uc11c \\(P_n(x)\\)\uc758 \uac12\uc744 \\(f(x)\\)\uc758 \uadfc\uc0bf\uac12\uc73c\ub85c \uc0ac\uc6a9\ud558\ub824\uba74 \ub450 \uac12\uc774 \uc5bc\ub9c8\ub098 \uac00\uae4c\uc6b4\uc9c0\ub97c \uac00\ub2a0\ud560 \uc218 \uc788\ub294 \uacf5\uc2dd\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.9. (\ud14c\uc77c\ub7ec \uc815\ub9ac)<\/span><\/p>\n<p>\\(a&lt;b\\)\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \\(C^n\\)\uc774\uba70 \\((a,b)\\)\uc5d0\uc11c \\((n+1)\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[<br \/>\nf(b)=\\sum_{k=0}^{n}\\frac{f^{(k)}(a)}{k!}(b-a)^k<br \/>\n+\\frac{f^{(n+1)}(c)}{(n+1)!}(b-a)^{n+1}.<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \uc6b0\ubcc0\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc744 <span class=\"defined\">\ub77c\uadf8\ub791\uc8fc \ub098\uba38\uc9c0<\/span>(Lagrange remainder)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744<br \/>\n\\[<br \/>\nP_n(x)=\\sum_{k=0}^{n}\\frac{f^{(k)}(a)}{k!}(x-a)^k<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \ub098\uba38\uc9c0 \\(R_n(x)=f(x)-P_n(x)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nf(b)=P_n(b)+M(b-a)^{n+1}<br \/>\n\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc0c1\uc218 \\(M\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \ud568\uc218<br \/>\n\\[<br \/>\ng(t)=f(t)-P_n(t)-M(t-a)^{n+1},\\quad t\\in[a,b]<br \/>\n\\]<br \/>\n\ub97c \uc815\uc758\ud558\uc790. \uadf8\ub7ec\uba74 \\(g(a)=g(b)=0\\)\uc774\uace0, \\(g\\)\ub294 \\([a,b]\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\((a,b)\\)\uc5d0\uc11c \\((n+1)\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\ub610\ud55c \\(k=0,1,\\ldots,n\\)\uc5d0 \ub300\ud558\uc5ec \\(g^{(k)}(a)=0\\)\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(P_n^{(k)}(a)=f^{(k)}(a)\\)\uc774\uace0 \\((t-a)^{n+1}\\)\uc758 \\(k\\)\uacc4\ub3c4\ud568\uc218\ub294 \\(t=a\\)\uc5d0\uc11c \\(0\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \ub864\uc758 \uc815\ub9ac\ub97c \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uc790. \\(g(a)=g(b)=0\\)\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(c_1\\in(a,b)\\)\uc5d0\uc11c \\(g'(c_1)=0\\)\uc774\ub2e4. \uc774\uc5b4\uc11c \\(g'(a)=g'(c_1)=0\\)\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(c_2\\in(a,c_1)\\)\uc5d0\uc11c \\(g&#8221;(c_2)=0\\)\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \uacc4\uc18d\ud558\uba74 \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0\uc11c \\(g^{(n+1)}(c)=0\\)\uc774\ub2e4.<\/p>\n<p>\uadf8\ub7f0\ub370<br \/>\n\\[<br \/>\ng^{(n+1)}(t)=f^{(n+1)}(t)-M(n+1)!<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nM=\\frac{f^{(n+1)}(c)}{(n+1)!}.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \ubc14\ub77c\ub294 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud14c\uc77c\ub7ec \uc815\ub9ac\uc758 \ub098\uba38\uc9c0\ub294 \ub77c\uadf8\ub791\uc8fc \ub098\uba38\uc9c0 \uc678\uc5d0 \uc5ec\ub7ec \ud615\ud0dc\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \uac19\uc740 \uac00\uc815 \uc544\ub798 \uc801\ub2f9\ud55c \\(c\\in(a,b)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nR_n=\\frac{f^{(n+1)}(c)}{n!}(b-c)^n(b-a)<br \/>\n\\]<br \/>\n\ub85c \ub098\ud0c0\ub0b4\ub294 <span class=\"defined\">\ucf54\uc2dc \ub098\uba38\uc9c0<\/span>(Cauchy remainder)\uac00 \uc788\ub2e4. \uc5ec\uae30\uc11c\ub294 \uc774 \uc2dd\uc744 \uc99d\uba85 \uc5c6\uc774 \uae30\ub85d\ud55c\ub2e4. \uc801\ubd84\uc744 \uc815\uc758\ud55c \ub4a4\uc5d0\ub294 \ub098\uba38\uc9c0\ub97c \uc801\ubd84\uc73c\ub85c \ub098\ud0c0\ub0b4\ub294 \uacf5\uc2dd\ub3c4 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<h3>\ud568\uc218\uc758 \uadf9\uac12\uacfc \ubcfc\ub85d\uc131<\/h3>\n<p>\ud568\uc218 \\(f\\colon E\\to\\mathbb R\\)\uc640 \\(c\\in E\\)\ub97c \uc0dd\uac01\ud558\uc790. \uc5b4\ub5a4 \\(\\delta&gt;0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in E\\cap B(c,\\delta)\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)\\le f(c)\\)\uc774\uba74 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c <span class=\"defined\">\uad6d\uc18c\uadf9\ub313\uac12<\/span>(local maximum)\uc744 \uac00\uc9c4\ub2e4\uace0 \ud55c\ub2e4. <span class=\"defined\">\uad6d\uc18c\uadf9\uc19f\uac12<\/span>(local minimum)\ub3c4 \ubd80\ub4f1\ud638\ub97c \ubc18\ub300\ub85c \ud558\uc5ec \uc815\uc758\ud55c\ub2e4. \uad6d\uc18c\uadf9\ub313\uac12\uacfc \uad6d\uc18c\uadf9\uc19f\uac12\uc744 \uac04\ub2e8\ud788 <span class=\"defined\">\uadf9\ub313\uac12<\/span>\uacfc <span class=\"defined\">\uadf9\uc19f\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud558\uba70, \ub458\uc744 \ud1b5\ud2c0\uc5b4 <span class=\"defined\">\uad6d\uc18c\uadf9\uac12<\/span> \ub610\ub294 <span class=\"defined\">\uadf9\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uadf9\uac12\uc744 \ud310\uc815\ud558\ub294 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 5.10. (\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc5b4\ub5a4 \\(\\delta&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\((c-\\delta,c+\\delta)\\subseteq I\\)\uc774\uace0, \\(c-\\delta&lt;x&lt;c\\)\uc774\uba74 \\(f'(x)&gt;0\\), \\(c&lt;x&lt;c+\\delta\\)\uc774\uba74 \\(f'(x)&lt;0\\)\uc77c \ub54c \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\uc5b4\ub5a4 \\(\\delta&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\((c-\\delta,c+\\delta)\\subseteq I\\)\uc774\uace0, \\(c-\\delta&lt;x&lt;c\\)\uc774\uba74 \\(f'(x)&lt;0\\), \\(c&lt;x&lt;c+\\delta\\)\uc774\uba74 \\(f'(x)&gt;0\\)\uc77c \ub54c \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\uc758 \uacbd\uc6b0 \uc815\ub9ac 5.7\uc5d0 \uc758\ud574 \\(f\\)\ub294 \\(c\\)\uc758 \uc67c\ucabd\uc5d0\uc11c\ub294 \uc21c\uc99d\uac00\ud558\uace0 \uc624\ub978\ucabd\uc5d0\uc11c\ub294 \uc21c\uac10\uc18c\ud55c\ub2e4. \ub530\ub77c\uc11c \ucda9\ubd84\ud788 \\(c\\)\uc5d0 \uac00\uae4c\uc6b4 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)\\le f(c)\\)\uc774\ub2e4. (2)\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud55c\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 5.11. (\uadf9\uac12\uc5d0 \ub300\ud55c \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(c\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f'(c)=0\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f&#8221;(c)&gt;0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(f&#8221;(c)&lt;0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n<li>\\(f&#8221;(c)=0\\)\uc778 \uac83\ub9cc\uc73c\ub85c\ub294 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\ub294\uc9c0 \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f&#8221;(c)&gt;0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc815\uc758\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n\\lim_{x\\to c}\\frac{f'(x)-f'(c)}{x-c}<br \/>\n=<br \/>\n\\lim_{x\\to c}\\frac{f'(x)}{x-c}<br \/>\n=f&#8221;(c)&gt;0.<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \ucda9\ubd84\ud788 \\(c\\)\uc5d0 \uac00\uae4c\uc6b4 \\(x\\neq c\\)\uc5d0 \ub300\ud558\uc5ec \\(f'(x)\/(x-c)&gt;0\\)\uc774\ub2e4. \uc989 \\(x&lt;c\\)\uc774\uba74 \\(f'(x)&lt;0\\), \\(x&gt;c\\)\uc774\uba74 \\(f'(x)&gt;0\\)\uc774\ubbc0\ub85c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc5d0 \uc758\ud574 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(f&#8221;(c)&lt;0\\)\uc778 \uacbd\uc6b0\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uadf9\ub313\uac12\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\ub9c8\uc9c0\ub9c9 \ud56d\uc740 \\(f(x)=x^4\\), \\(f(x)=-x^4\\), \\(f(x)=x^3\\)\uc744 \\(c=0\\)\uc5d0\uc11c \ube44\uad50\ud558\uba74 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.9.<\/span><br \/>\n\ub2e4\uc74c \uc138 \ud568\uc218\uc5d0 \ub300\ud558\uc5ec \\(x=0\\)\uc774 \uadf9\uac12\uc810\uc778\uc9c0 \ud310\uc815\ud558\uace0, \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\ub9cc\uc73c\ub85c \uacb0\ub860\uc744 \ub0bc \uc218 \uc788\ub294\uc9c0\ub3c4 \uc124\uba85\ud558\uc2dc\uc624.<br \/>\n\\[<br \/>\nf_1(x)=x^4,\\quad f_2(x)=-x^4,\\quad f_3(x)=x^3.<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.10.<\/span><br \/>\n\ud568\uc218 \\(f(x)=x^3-3x+1\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uace0, \uac01 \uadf9\uac12\uc774 \uadf9\ub313\uac12\uc778\uc9c0 \uadf9\uc19f\uac12\uc778\uc9c0 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<p>\uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\ubcfc\ub85d<\/span>\ud558\ub2e4(convex)\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(x,y\\in I\\)\uc640 \\(t\\in[0,1]\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<br \/>\n\\[<br \/>\nf(tx+(1-t)y)\\leq tf(x)+(1-t)f(y).\\tag{5.1}<br \/>\n\\]<br \/>\n\ub9cc\uc57d \ubd80\ub4f1\ud638\ub97c \ubc18\ub300\ub85c \ubc14\uafbc \uc870\uac74\uc774 \uc131\ub9bd\ud558\uba74 \u201c\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c <span class=\"defined\">\uc624\ubaa9<\/span>\ud558\ub2e4(concave)\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ubd80\ub4f1\uc2dd (5.1)\uc740 \\(I\\)\uc758 \uc784\uc758\uc758 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec, \\(f\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \ub450 \uc810 \\((x,f(x))\\)\uc640 \\((y,f(y))\\)\ub97c \uc774\uc740 \uc120\ubd84\uc774 \\(f\\)\uc758 \uadf8\ub798\ud504\ubcf4\ub2e4 \uc544\ub798\ucabd\uc5d0 \uc788\uc9c0 \uc54a\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.12.<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(f&#8221;\\geq0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(f&#8221;\\leq0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc624\ubaa9\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n(1)\ub9cc \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. (2)\ub294 (1)\uc744 \\(-f\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4.<\/p>\n<p>\\(I\\)\uc5d0\uc11c \\(f&#8221;\\ge0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(x&lt;y\\)\uc778 \\(I\\)\uc758 \ub450 \uc810\uacfc \\(t\\in(0,1)\\)\uc744 \ud0dd\ud558\uace0<br \/>\n\\[<br \/>\nz=tx+(1-t)y<br \/>\n\\]<br \/>\n\ub77c\uace0 \ub193\uc790. \uadf8\ub7ec\uba74 \\(x&lt;z&lt;y\\)\uc774\ub2e4. \\(f&#8221;\\ge0\\)\uc774\ubbc0\ub85c \uc815\ub9ac 5.7\uc5d0 \uc758\ud574 \\(f&#8217;\\)\uc740 \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4. \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(\\xi_1\\in(x,z)\\), \\(\\xi_2\\in(z,y)\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\n\\frac{f(z)-f(x)}{z-x}=f'(\\xi_1),<br \/>\n\\quad<br \/>\n\\frac{f(y)-f(z)}{y-z}=f'(\\xi_2).<br \/>\n\\]<br \/>\n\\(\\xi_1&lt;\\xi_2\\)\uc774\ubbc0\ub85c \\(f'(\\xi_1)\\le f'(\\xi_2)\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n(y-x)f(z)\\le(y-z)f(x)+(z-x)f(y).<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(y-z=t(y-x)\\), \\(z-x=(1-t)(y-x)\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nf(tx+(1-t)y)\\le tf(x)+(1-t)f(y).<br \/>\n\\]<br \/>\n\\(x=y\\)\uc774\uac70\ub098 \\(t=0,1\\)\uc778 \uacbd\uc6b0\ub294 \uc790\uba85\ud558\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.11.<\/span><br \/>\n\\(I\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f4\ub9b0\uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.12.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0\uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f&#8217;\\)\uc774 \\(I\\)\uc5d0\uc11c \ub2e8\uc870\uc99d\uac00\ud568\uc218\uc778 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.13. (\uc60c\uc13c \ubd80\ub4f1\uc2dd)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud55c \ud568\uc218\uc774\uace0 \\(i=1,2,\\ldots,n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_i\\in I\\), \\(\\lambda_i\\geq0\\)\uc774\uba70 \\(\\sum_{i=1}^{n}\\lambda_i=1\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[<br \/>\nf\\left(\\sum_{i=1}^{n}\\lambda_i x_i\\right)<br \/>\n\\leq<br \/>\n\\sum_{i=1}^{n}\\lambda_i f(x_i).<br \/>\n\\]<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c \uc99d\uba85\ud558\uc790. \\(n=1\\)\uc77c \ub54c\ub294 \uc790\uba85\ud558\uace0, \\(n=2\\)\uc77c \ub54c\ub294 \ubcfc\ub85d\ud568\uc218\uc758 \uc815\uc758\uc774\ub2e4.<\/p>\n<p>\\(n=k\\)\uc77c \ub54c \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0 \\(n=k+1\\)\uc77c \ub54c\ub97c \ubcf4\uc774\uc790. \\(\\lambda_1,\\ldots,\\lambda_{k+1}\\geq0\\), \\(\\sum_{i=1}^{k+1}\\lambda_i=1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\lambda_{k+1}=1\\)\uc778 \uacbd\uc6b0\ub294 \uc790\uba85\ud558\ubbc0\ub85c \\(\\lambda_{k+1}&lt;1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(i=1,\\ldots,k\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\mu_i=\\frac{\\lambda_i}{1-\\lambda_{k+1}}<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(\\sum_{i=1}^k\\mu_i=1\\)\uc774\ub2e4. \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\nf\\left(\\sum_{i=1}^k\\mu_i x_i\\right)<br \/>\n\\leq<br \/>\n\\sum_{i=1}^k\\mu_i f(x_i).<br \/>\n\\]<\/p>\n<p>\uc774\uc81c \\(y=\\sum_{i=1}^k\\mu_i x_i\\)\ub77c\uace0 \ub193\uc73c\uba74<br \/>\n\\[<br \/>\n\\sum_{i=1}^{k+1}\\lambda_i x_i<br \/>\n=(1-\\lambda_{k+1})y+\\lambda_{k+1}x_{k+1}.<br \/>\n\\]<br \/>\n\ubcfc\ub85d\ud568\uc218\uc758 \uc815\uc758\uc640 \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud574<br \/>\n\\[<br \/>\n\\begin{aligned}<br \/>\nf\\left(\\sum_{i=1}^{k+1}\\lambda_i x_i\\right)<br \/>\n&#038;\\leq(1-\\lambda_{k+1})f(y)+\\lambda_{k+1}f(x_{k+1})\\\\<br \/>\n&#038;\\leq(1-\\lambda_{k+1})\\sum_{i=1}^k\\mu_i f(x_i)+\\lambda_{k+1}f(x_{k+1})\\\\<br \/>\n&#038;=\\sum_{i=1}^{k+1}\\lambda_i f(x_i).<br \/>\n\\end{aligned}<br \/>\n\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\uc704 \uc815\ub9ac\ub294 \uc801\ubd84\uc73c\ub85c \ud45c\ud604\ud560 \uc218\ub3c4 \uc788\ub2e4. \ub4a4\uc5d0\uc11c \uc801\ubd84\uc73c\ub85c \ud45c\ud604\ud55c \uc60c\uc13c \ubd80\ub4f1\uc2dd\uc744 \ubb38\uc81c\ub85c \ub2e4\ub8ec\ub2e4.<!-- TODO: problem:jenseninequalityintegral\uc758 \ucd5c\uc885 \ubb38\uc81c \ubc88\ud638 \ubc0f \uc628\ub77c\uc778 \ub9c1\ud06c \ud655\uc778 --><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \\(c\\)\uc758 \ud55c\ucabd\uc5d0\uc11c\ub294 \\(f\\)\uac00 \ubcfc\ub85d\ud558\uace0 \ub2e4\ub978 \ucabd\uc5d0\uc11c\ub294 \uc624\ubaa9\ud558\uc5ec \\(c\\)\ub97c \uc9c0\ub098\uba74\uc11c \ubcfc\ub85d\uc131\uc774 \ubc14\ub00c\uba74, \\((c,f(c))\\)\ub97c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 <span class=\"defined\">\ubcc0\uace1\uc810<\/span>(inflection point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(f&#8221;\\)\uc774 \\(c\\)\uc758 \uc591\ucabd\uc5d0\uc11c \uc874\uc7ac\ud558\uace0 \\(c\\)\ub97c \uc9c0\ub098\uba74\uc11c \\(f&#8221;\\)\uc758 \ubd80\ud638\uac00 \ubc14\ub00c\uba74 \\(c\\)\ub294 \ubcc0\uace1\uc810\uc774\ub2e4.<\/p>\n<h3>\ubbf8\ubd84\uc758 \uc5ec\ub7ec \uc751\uc6a9<\/h3>\n<p>\ud568\uc218\uac00 \uad6c\uac04\uc5d0\uc11c \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \uadf8 \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.14. (\uc5ed\ud568\uc218 \uc815\ub9ac (1\ucc28\uc6d0))<\/span><\/p>\n<p>\ud568\uc218 \\(f\\colon I\\to\\mathbb R\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc21c\uc99d\uac00\ud558\uac70\ub098 \uc21c\uac10\uc18c\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f\\colon I\\to f(I)\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0 \uc5ed\ud568\uc218<br \/>\n\\[<br \/>\nf^{-1}\\colon f(I)\\to I<br \/>\n\\]<br \/>\n\ub294 \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \\(f'(c)\\neq0\\)\uc774\uba74 \\(f^{-1}\\)\uc740 \\(f(c)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0<br \/>\n\\[<br \/>\n(f^{-1})'(f(c))=\\frac1{f'(c)}<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\\(f\\)\uac00 \uc21c\uc99d\uac00\ud558\ub294 \uacbd\uc6b0\ub9cc \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. \uc5f0\uc18d\uc131\uacfc \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(f(I)\\)\ub294 \uad6c\uac04\uc774\uace0, \uc21c\uc99d\uac00\uc131\uc5d0 \uc758\ud574 \\(f\\colon I\\to f(I)\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4.<\/p>\n<p>\uba3c\uc800 \uc5ed\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc744 \ubcf4\uc774\uc790. \uc784\uc758\uc758 \\(c\\in I\\)\ub97c \ud0dd\ud558\uace0 \\(d=f(c)\\)\ub77c\uace0 \ud558\uc790. \\(c\\)\uac00 \ub0b4\uc810\uc774\uba74 \ucda9\ubd84\ud788 \uc791\uc740 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(c-\\varepsilon,c+\\varepsilon\\in I\\)\uc774\uace0<br \/>\n\\[<br \/>\nf(c-\\varepsilon)&lt;d&lt;f(c+\\varepsilon).<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\eta=\\min\\{d-f(c-\\varepsilon),\\,f(c+\\varepsilon)-d\\}&gt;0<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(y\\in f(I)\\), \\(|y-d|&lt;\\eta\\)\uc77c \ub54c<br \/>\n\\[<br \/>\nc-\\varepsilon&lt;f^{-1}(y)&lt;c+\\varepsilon.<br \/>\n\\]<br \/>\n\ub05d\uc810\uc5d0\uc11c\ub294 \ud55c\ubc29\ud5a5\uc73c\ub85c \uac19\uc740 \ub17c\uc99d\uc744 \uc801\uc6a9\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \\(f^{-1}\\)\uc740 \\(f(I)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \uc21c\uac10\uc18c\uc778 \uacbd\uc6b0\ub3c4 \ubd80\ub4f1\ud638\ub9cc \ubc14\uafb8\uba74 \ub41c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(c\\)\uac00 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \\(f'(c)\\neq0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(y\\to d=f(c)\\), \\(y\\in f(I)\\), \\(y\\neq d\\)\uc77c \ub54c \\(x=f^{-1}(y)\\)\ub77c\uace0 \ub193\uc73c\uba74 \uc5ed\ud568\uc218\uc758 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(x\\to c\\)\uc774\uace0 \\(x\\neq c\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\frac{f^{-1}(y)-f^{-1}(d)}{y-d}<br \/>\n=<br \/>\n\\frac{x-c}{f(x)-f(c)}<br \/>\n=<br \/>\n\\frac1{\\dfrac{f(x)-f(c)}{x-c}}<br \/>\n\\longrightarrow\\frac1{f'(c)}.<br \/>\n\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \ubc14\ub77c\ub294 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.13.<\/span><br \/>\n\\(r\\)\uc774 \\(0\\) \uc544\ub2cc \uc720\ub9ac\uc218\uc774\uace0 \\(x&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)=x^r\\)\uc774\ub77c\uace0 \ud560 \ub54c \\(f'(x)=rx^{r-1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.14.<\/span><br \/>\n\ubbf8\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624. (<a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ubb38\uc81c 4.7<\/a>\uc744 \ucc38\uc870\ud558\uc2dc\uc624.)<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(x)=e^x\\)<\/li>\n<li>\\(f(x)=\\ln x\\) (\ub2e8, \\(x&gt;0\\).)<\/li>\n<li>\\(f(x)=a^x\\) (\ub2e8, \\(a&gt;0\\), \\(a\\ne1\\).)<\/li>\n<li>\\(f(x)=\\log_a x\\) (\ub2e8, \\(a&gt;0\\), \\(a\\ne1\\), \\(x&gt;0\\).)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.15.<\/span><br \/>\n\ud568\uc218 \\(f(x)=e^x\\)\uc758 \\(0\\)\uc5d0\uc11c\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(x=1\\)\uc5d0\uc11c \\(e\\)\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud560 \ub54c, \uc624\ucc28\uac00<br \/>\n\\[<br \/>\n\\frac{3}{(n+1)!}<br \/>\n\\]<br \/>\n\ubcf4\ub2e4 \uc791\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624. <a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\ubb38\uc81c 1.16<\/a>\uc5d0\uc11c \uc5bb\uc740 \\(e&lt;3\\)\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.16.<\/span><br \/>\n\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n1+x\\le e^x<br \/>\n\\]<br \/>\n\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.17.<\/span><br \/>\n\\(a&gt;0\\), \\(a\\ne1\\)\uc774\uace0 \\(x\\)\uac00 \uc2e4\uc218\uc77c \ub54c, \\(a^x=e^{x\\ln a}\\)\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.18.<\/span><br \/>\n\\(\\alpha\\)\uac00 \ubb34\ub9ac\uc218\uc774\uace0 \\(x&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)=x^\\alpha\\)\ub77c\uace0 \ud560 \ub54c \\(f'(x)=\\alpha x^{\\alpha-1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.19.<\/span><br \/>\n\uc2e4\uc218 \\(r\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(x&gt;-1\\)\uc774\uace0 \\(r\\ge1\\)\uc77c \ub54c \\((1+x)^r\\ge1+rx\\)\uc774\ub2e4.<\/li>\n<li>\\(x&gt;-1\\)\uc774\uace0 \\(0\\le r\\le1\\)\uc77c \ub54c \\((1+x)^r\\le1+rx\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\\(x=-1\\)\uc774\uace0 \\(r&gt;0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc815\uc758\ub418\ub294 \ubc94\uc704\uc5d0\uc11c \uac19\uc740 \ubd80\ub4f1\uc2dd\uc774 \uc9c1\uc811 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud558\uc2dc\uc624. \uc774\ub97c \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd\uc758 \uc2e4\uc218 \uc9c0\uc218 \ud615\ud0dc\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.20.<\/span><br \/>\n\uc5ed\ud568\uc218 \uc815\ub9ac(\uc815\ub9ac 5.14)\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(f(x)=\\sin^{-1}x\\), \uc5ec\uae30\uc11c \\(\\sin\\)\uc740 \\((-\\pi\/2,\\pi\/2)\\)\uc5d0 \uc81c\ud55c\ud55c\ub2e4. \\((-1&lt;x&lt;1)\\)<\/li>\n<li>\\(f(x)=\\cos^{-1}x\\), \uc5ec\uae30\uc11c \\(\\cos\\)\uc740 \\((0,\\pi)\\)\uc5d0 \uc81c\ud55c\ud55c\ub2e4. \\((-1&lt;x&lt;1)\\)<\/li>\n<li>\\(f(x)=\\tan^{-1}x\\), \uc5ec\uae30\uc11c \\(\\tan\\)\uc740 \\((-\\pi\/2,\\pi\/2)\\)\uc5d0 \uc81c\ud55c\ud55c\ub2e4. \\((x\\in\\mathbb R)\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.21.<\/span><br \/>\n\ud568\uc218<br \/>\n\\[<br \/>\nf(x)=|\\sin x|<br \/>\n\\]<br \/>\n\uac00 \\(\\mathbb R\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0, \uc815\ud655\ud788 \\(x=k\\pi\\) (\\(k\\in\\mathbb Z\\))\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ud568\uc218 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub354\ub77c\ub3c4 \\(f&#8217;\\)\uc740 \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uadf8\ub7fc\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc0ac\uc787\uac12 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p><span class=\"definition\">\uc815\ub9ac 5.15. (\ub2e4\ub974\ubd80\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(a&lt;b\\)\uc774\uace0 \\(f\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f'(a)\\neq f'(b)\\)\uc774\uba74, \\(f'(a)\\)\uc640 \\(f'(b)\\) \uc0ac\uc774\uc758 \ubaa8\ub4e0 \uac12\uc774 \\(f&#8217;\\)\uc758 \uce58\uc5ed\uc5d0 \ud3ec\ud568\ub41c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f'(a)&lt;k&lt;f'(b)\\)\ub77c\uace0 \ud558\uc790. \\(g(x)=f(x)-kx\\)\ub77c\uace0 \ud558\uba74 \\(g'(x)=f'(x)-k\\)\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\ng'(a)=f'(a)-k&lt;0,\\quad g'(b)=f'(b)-k&gt;0.<br \/>\n\\]<br \/>\n\\(g'(a)&lt;0\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \uc791\uc740 \\(h&gt;0\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\frac{g(a+h)-g(a)}{h}&lt;0,<br \/>\n\\]<br \/>\n\ub530\ub77c\uc11c \\(g(a+h)&lt;g(a)\\)\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(g'(b)&gt;0\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \uc791\uc740 \\(h&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(g(b-h)&lt;g(b)\\)\uc774\ub2e4. \\(g\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uc9d1\ud569\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(g\\)\ub294 \\(a\\)\ub098 \\(b\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c \uc5b4\ub5a4 \\(c\\in(a,b)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\ub294\ub2e4. \ud398\ub974\ub9c8\uc758 \uc815\ub9ac(\uc815\ub9ac 5.3)\uc5d0 \uc758\ud574 \\(g'(c)=0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f'(c)=k\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<p>\ud568\uc218\uc758 \uadf9\ud55c\uc774 \ubd80\uc815\ud615\uc77c \ub54c, \ub3c4\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \uc720\uc6a9\ud55c \uacf5\uc2dd\uc774 \uc788\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p class=\"marginbottomhalf\"><span class=\"definition\">\uc815\ub9ac 5.16. (\ub85c\ud53c\ud0c8\uc758 \uc815\ub9ac)<\/span><\/p>\n<p>\\(a\\in\\mathbb R\\)\uc774\uace0 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\(a\\)\uc758 \uad6c\uba4d\ub6ab\ub9b0 \uadfc\ubc29\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \uadf8 \uadfc\ubc29\uc5d0\uc11c \\(g'(x)\\neq0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c \ub450 \uacbd\uc6b0 \uc911 \ud558\ub098\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"parenthesis\">\n<li>\n\\[<br \/>\n\\lim_{x\\to a}f(x)=\\lim_{x\\to a}g(x)=0.\\tag{5.2}<br \/>\n\\]\n<\/li>\n<li>\n\\[<br \/>\n\\lim_{x\\to a}|f(x)|=\\lim_{x\\to a}|g(x)|=\\infty.<br \/>\n\\]\n<\/li>\n<\/ol>\n<p>\ub610\ud55c<br \/>\n\\[<br \/>\n\\lim_{x\\to a}\\frac{f'(x)}{g'(x)}=L<br \/>\n\\]<br \/>\n\uc774 \uc874\uc7ac\ud55c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[<br \/>\n\\lim_{x\\to a}\\frac{f(x)}{g(x)}=L.\\tag{5.3}<br \/>\n\\]<br \/>\n\uc5ec\uae30\uc11c \\(L\\)\uc740 \uc720\ud55c\ud55c \uc2e4\uc218\ubfd0 \uc544\ub2c8\ub77c \\(+\\infty\\) \ub610\ub294 \\(-\\infty\\)\uc77c \uc218\ub3c4 \uc788\ub2e4. \uac19\uc740 \uba85\uc81c\ub294 \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc5d0 \ub300\ud574\uc11c\ub3c4 \uc131\ub9bd\ud558\uba70, \\(x\\to\\pm\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\uc99d\uba85<\/span><br \/>\n\uba3c\uc800 \\(0\/0\\) \uaf34\uc758 \uacbd\uc6b0\ub97c \ubcf4\uc774\uc790. \\(f(a)=g(a)=0\\)\uc73c\ub85c \uc815\uc758\ud558\uba74 \ub450 \ud568\uc218\ub294 \\(a\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \ub41c\ub2e4. \\(x\\neq a\\)\uac00 \\(a\\)\uc5d0 \ucda9\ubd84\ud788 \uac00\uae4c\uc6b8 \ub54c \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0\uc11c \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74 \uc5b4\ub5a4 \\(c_x\\)\uac00 \\(a\\)\uc640 \\(x\\) \uc0ac\uc774\uc5d0 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[<br \/>\n\\frac{f(x)}{g(x)}<br \/>\n=<br \/>\n\\frac{f(x)-f(a)}{g(x)-g(a)}<br \/>\n=<br \/>\n\\frac{f'(c_x)}{g'(c_x)}<br \/>\n\\]<br \/>\n\uc774\ub2e4. \uc774\ub54c \\(c_x\\to a\\)\uc774\ubbc0\ub85c \uc624\ub978\ucabd\uc740 \\(L\\)\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \\(\\infty\/\\infty\\) \uaf34\uc758 \uacbd\uc6b0\ub97c \ud55c\ubc29\ud5a5 \uadf9\ud55c\uc5d0\uc11c \uc124\uba85\ud558\uc790. \uc720\ud55c\ud55c \\(L\\)\uc758 \uacbd\uc6b0, \uc784\uc758\uc758 \\(\\varepsilon&gt;0\\)\uc5d0 \ub300\ud558\uc5ec \\(a\\)\uc5d0 \ucda9\ubd84\ud788 \uac00\uae4c\uc6b4 \ud55c\ucabd \uadfc\ubc29\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\left|\\frac{f'(x)}{g'(x)}-L\\right|&lt;\\varepsilon<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uadf8 \uadfc\ubc29 \uc548\uc758 \uc810 \\(x_0\\)\ub97c \ud558\ub098 \uace0\uc815\ud558\uace0, \\(x\\)\ub97c \\(a\\)\uc5d0 \ub354 \uac00\uae4c\uc6b4 \uc810\uc73c\ub85c \ud0dd\ud55c\ub2e4. \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(x\\)\uc640 \\(x_0\\) \uc0ac\uc774\uc758 \uc5b4\ub5a4 \\(c_x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nQ_x:=\\frac{f(x)-f(x_0)}{g(x)-g(x_0)}<br \/>\n=\\frac{f'(c_x)}{g'(c_x)}<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(|Q_x-L|&lt;\\varepsilon\\)\uc774\ub2e4. \ud55c\ud3b8<br \/>\n\\[<br \/>\n\\frac{f(x)}{g(x)}<br \/>\n=<br \/>\nQ_x\\left(1-\\frac{g(x_0)}{g(x)}\\right)<br \/>\n+<br \/>\n\\frac{f(x_0)}{g(x)}.<br \/>\n\\]<br \/>\n\\(|g(x)|\\to\\infty\\)\uc774\ubbc0\ub85c \ub4a4\uc758 \ub450 \ubcf4\uc815\ud56d\uc740 \uc0ac\ub77c\uc9c0\uace0, \ub530\ub77c\uc11c \\(f(x)\/g(x)\\to L\\)\uc774\ub2e4. \\(L=\\pm\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub3c4 \uc704 \ub17c\uc99d\uc5d0\uc11c \\(|Q_x|\\)\ub97c \uc784\uc758\ub85c \ud06c\uac8c \ub9cc\ub4dc\ub294 \ubc29\uc2dd\uc73c\ub85c \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \ubc18\ub300\ucabd \ud55c\ubc29\ud5a5 \uadf9\ud55c\ub3c4 \ub3d9\uc77c\ud558\ub2e4.<\/p>\n<p>\\(x\\to+\\infty\\) \ub610\ub294 \\(x\\to-\\infty\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \uc704\uc758 \\(\\infty\/\\infty\\) \ub17c\uc99d\uc744 \ucda9\ubd84\ud788 \ud070 \\(|x|\\)\uc5d0\uc11c \uc9c1\uc811 \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4. \\(0\/0\\) \uaf34\uc740 \\(F(t)=f(1\/t)\\), \\(G(t)=g(1\/t)\\)\ub85c \ub450\uc5b4 \\(t\\to0+\\) \ub610\ub294 \\(t\\to0-\\)\uc778 \uacbd\uc6b0\ub85c \ud658\uc6d0\ud560 \uc218 \uc788\uace0,<br \/>\n\\[<br \/>\n\\frac{F'(t)}{G'(t)}<br \/>\n=<br \/>\n\\frac{f'(1\/t)}{g'(1\/t)}<br \/>\n\\]<br \/>\n\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 5.22.<\/span><br \/>\n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle\\lim_{x\\to\\infty}\\frac{x^2}{e^x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0+}x\\ln x\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to0+}(1+3x)^{1\/x}\\)<\/li>\n<li>\\(\\displaystyle\\lim_{x\\to1+}(\\ln x)^{1-x}\\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 5.23.<\/span><br \/>\n\ud568\uc218 \\(f\\)\uac00 \\(\\mathbb R\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(A\\)\uac00 \uc2e4\uc218\uc774\uba70 \\(\\displaystyle\\lim_{x\\to\\infty}(f(x)+f'(x))=A\\)\uc774\uba74 \\(\\displaystyle\\lim_{x\\to\\infty}f(x)=A\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n<ol class=\"contentboxorderedlist\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\uac70\ub9ac\uacf5\uac04<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n<li class=\"contentboxthis\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\">\uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc911\uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\">\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc55e \uc7a5\uc5d0\uc11c\ub294 \uc2e4\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131\uc744 \uc0b4\ud3b4\ubcf4\uc558\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \uc815\uc758\ud558\uace0, \ud3c9\uade0\uac12 \uc815\ub9ac, \ud14c\uc77c\ub7ec \uc815\ub9ac, \ubcfc\ub85d\uc131 \ub4f1 \uc77c\ubcc0\uc218 \ubbf8\ubd84\ubc95\uc758 \uc8fc\uc694 \uacb0\uacfc\ub97c \ub2e4\ub8ec\ub2e4. \ubbf8\ubd84 \uac00\ub2a5\uc131 \\(X\\)\uac00 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(c\\in X\\cap X&#8217;\\)\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f\\colon X\\to\\mathbb R\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4(differentiable)\ub294 \uac83\uc740 \\(X\\) \uc548\uc5d0\uc11c \ub2e4\uc74c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \\( f'(c)=\\lim_{x\\to c}\\frac{f(x)-f(c)}{x-c}. \\) \uc774 \uadf9\ud55c\uac12 \\(f'(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218(derivative)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(h=x-c\\)\ub85c \ub193\uc73c\uba74,&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":105,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9486","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=9486"}],"version-history":[{"count":12,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486\/revisions"}],"predecessor-version":[{"id":10109,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486\/revisions\/10109"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}