\\(D\\)\uac00 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790.
\n\\[D = \\left\\{ (x,\\,y) \\,\\vert\\, x^2 + y^2 \\le 9 \\right\\}\\]
\n\uadf8\ub9ac\uace0 \ud568\uc218 \\(f\\)\uac00 \\(D\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\uc73c\uba70 \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ub450 \ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \\(D\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ub313\uac12\uacfc \uadf9\uc19f\uac12, \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n
\uadf8\ub7ec\ub098 \\(D\\)\uac00 \uc720\uacc4\uac00 \uc544\ub2c8\uac70\ub098 \ub2eb\ud78c\uc9d1\ud569\uc774 \uc544\ub2c8\ub77c\uba74 \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc5c6\ub2e4. \uc774\ub54c\uc5d0\ub294 \ud568\uc218\uc758 \ud2b9\uc9d5\uc5d0 \ub530\ub77c \ub2e4\uc591\ud55c \ubc29\ubc95\uc73c\ub85c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud574\uc57c \ud55c\ub2e4.<\/p>\n
\n\uc608\uc81c 1.<\/span> \n\ud480\uc774.<\/span> \uadf8\ub807\ub2e4\uba74 \\(f\\)\ub294 \\((2,\\,-1)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c8\uae4c? \uadf8\ub807\ub2e4. \uc77c\ub2e8 \\(f\\)\uc758 \uc784\uacc4\uc810\uc740 \\((2,\\,-1)\\) \ubfd0\uc774\ubbc0\ub85c \ud2b9\uc815\ud55c \ub2e4\ub978 \uc810\uc5d0\uc11c \\(f\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc774\uc81c \ub0a8\uc740 \uacbd\uc6b0\uc758 \uc218\ub294 \\((x,\\,y)\\)\uc758 \uc6c0\uc9c1\uc784\uc5d0 \ub530\ub77c \\(f\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uacbd\uc6b0\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc774\ub7ec\ud55c \uacbd\uc6b0 \ub610\ud55c \ubc1c\uc0dd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74 \uc810 \\((x,\\,y)\\)\uac00 \uc6d0\uc810\uc73c\ub85c\ubd80\ud130 \uba40\ub9ac \ub5a8\uc5b4\uc9c8\uc218\ub85d, \uc989 \\( | (x,\\,y)| \\,\\rightarrow\\, \\infty\\)\uc77c \ub54c \\(f(x,\\,y) \\,\\rightarrow \\,\\infty\\)\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \uc544\ub798\ub85c \uc720\uacc4\uc774\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\uc5c8\ub2e4\uba74 \\(f\\)\uac00 \uc74c\uc758 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub3c4\ub85d \\((x,\\,y)\\)\ub97c \uc6c0\uc9c1\uc77c \uc218 \uc788\uc5c8\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n \uc774\ub85c\uc368 \\(f\\)\uac00 \\((2,\\,-1)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4\ub294 \uc0ac\uc2e4\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n \ud55c\ud3b8 \\(x \\to \\infty ,\\) \\(y \\to \\infty\\)\uc77c \ub54c \\(f(x,\\,y) \\to \\infty\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \ucd5c\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. \uc815\uc758\uc5ed\uc774 \uc720\uacc4\uac00 \uc544\ub2cc \ub610 \ub2e4\ub978 \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n \n\uc608\uc81c 2.<\/span> \n\ud480\uc774.<\/span> \uc774\uc81c \\(f\\)\uac00 \\((1,\\,1)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\ubcc4\ud574 \ubcf4\uc790. \ub9cc\uc57d \\(x\\)\uac00 \ubb34\ud55c\ud788 \ucee4\uc9c0\uace0 \\(y\\)\ub294 \ubb34\ud55c\ud788 \uc791\uc544\uc9c4\ub2e4\uba74, \uc608\ucee8\ub300 \uc9c1\uc120 \\(y=-x\\)\ub97c \ub530\ub77c\uc11c \\(x \\,\\rightarrow \\,\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud55c\ub2e4\uba74 \ud55c\ud3b8 \\(x \\to \\infty ,\\) \\(y \\to \\infty\\)\uc77c \ub54c \\(f(x,\\,y) \\to \\infty\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \ucd5c\ub313\uac12\uc744 \uac16\uc9c0\ub3c4 \uc54a\ub294\ub2e4. \ub2e4\uc74c\uc73c\ub85c \uc815\uc758\uc5ed\uc774 \\(\\mathbb{R}^2\\) \uc804\uccb4\uac00 \uc544\ub2c8\uc9c0\ub9cc \uc720\uacc4\ub3c4 \uc544\ub2cc \uc608\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n \n\uc608\uc81c 3.<\/span> \n\ud480\uc774.<\/span> \uc774\uc81c \\(f\\)\uac00 \\(( \\frac{1}{2} ,\\,2)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\ubcc4\ud574 \ubcf4\uc790. \\(xy > 0\\)\uc774\ubbc0\ub85c \ud2b9\ud788 \\(2x – \\ln x\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac16\ub294 \uac83\uc740 \\(x = \\frac{1}{2}\\)\uc77c \ub54c\uc774\uace0, \\(xy – \\ln xy\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac16\ub294 \uac83\uc740 \\(xy = 1\\) \uc989 \\(y = \\frac{1}{x}\\)\uc77c \ub54c\uc774\ubbc0\ub85c, \\(f\\)\uac00 \\((\\frac{1}{2} ,\\, 2 )\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4\ub294 \uc0ac\uc2e4\uc774 \ub2e4\uc2dc \ud55c \ubc88 \ud655\uc778\ub41c\ub2e4.<\/p>\n \uc774\ub85c\uc368 \\(f\\)\uac00 \\((\\frac{1}{2} ,\\,2)\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4\ub294 \uc0ac\uc2e4\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n \ud55c\ud3b8 \ud480\uc774 \uacfc\uc815\uc5d0\uc11c \\((x,\\,y)\\)\uc758 \uc6c0\uc9c1\uc784\uc5d0 \ub530\ub77c \\(f(x,\\,y) \\to \\infty\\)\uc778 \uacbd\uc6b0\ub97c \ubcf4\uc558\uc73c\ubbc0\ub85c, \\(f\\)\ub294 \ucd5c\ub313\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4. <\/p>\n","protected":false},"excerpt":{"rendered":" \\(D\\)\uac00 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc73c\ub85c\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \\(D = \\left\\{ (x,\\,y) \\,\\vert\\, x^2 + y^2 \\le 9 \\right\\}\\) \uadf8\ub9ac\uace0 \ud568\uc218 \\(f\\)\uac00 \\(D\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc601\uc5ed\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\uc73c\uba70 \uc815\uc758\uc5ed\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ub450 \ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\uc81c \\(D\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ub313\uac12\uacfc \uadf9\uc19f\uac12, \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uad6c\ud560 \uc218 \uc788\ub2e4. \ub3c4\ud568\uc218 \ud310\uc815\ubc95: \\(D\\)\uc758 \ub0b4\ubd80\uc810 \uc911\uc5d0\uc11c \\(f_x (x,\\,y) = 0,\\) \\(f_y (x,\\,y)=0\\)\uc778 \uc810 \\((x,\\,y)\\)\ub97c \ubaa8\ub450 \uad6c\ud55c\ub2e4. \uc774\uacc4\ub3c4\ud568\uc218…<\/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":[],"class_list":["post-4630","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4630","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=4630"}],"version-history":[{"count":25,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4630\/revisions"}],"predecessor-version":[{"id":4659,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4630\/revisions\/4659"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4630"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4630"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\(\\mathbb{R}^2\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uac00
\n\\[f(x,\\,y) = 2x^2 + 3xy + 4y^2 – 5x + 2y\\]
\n\ub85c \uc815\uc758\ub418\uc5c8\uc744 \ub54c, \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n
\n\uba3c\uc800 \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uc790. \\(f\\)\uc758 \ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[\\begin{gather}
\nf_x (x,\\,y) = 3x+3y-5 \\\\[6pt]
\nf_y (x,\\,y) = 8y + 3x+2
\n\\end{gather}\\]
\n\uc774\ub2e4. \ub450 \ud3b8\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \\(0\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uc810\uc744 \uad6c\ud558\uba74 \\((x,\\,y) = (2,\\,-1)\\) \ubfd0\uc774\ub2e4. \uc774 \uc810\uc5d0\uc11c \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95\uc744 \uc774\uc6a9\ud558\uc790.
\n\\[f_{xx} (x,\\,y) = 4 ,\\quad f_{yy} (x,\\,y) = 8 ,\\quad f_{xy} (x,\\,y) = f_{yx} (x,\\,y) = 3\\]
\n\uc774\ubbc0\ub85c
\n\\[\\det H(f) = 4 \\times 8 – 3^2 = 23 > 0\\]
\n\uc774\ub2e4. \ub354\uc6b1\uc774 \\(f_{xx} (2,\\,-1) > 0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\((2,\\,-1)\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n
\n<\/span>\n<\/p>\n<\/div>\n
\n\\(\\mathbb{R}^2\\)\uc758 \\(xy \\ne 0\\)\uc778 \uc810\uc5d0\uc11c \ud568\uc218 \\(f\\)\uac00
\n\\[f(x,\\,y) = \\frac{1}{x} + xy + \\frac{1}{y}\\]
\n\ub85c \uc815\uc758\ub418\uc5c8\uc744 \ub54c, \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n
\n\uba3c\uc800 \\(f\\)\uc758 \ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[f_x (x,\\,y) = -\\frac{1}{x^2} + y ,\\quad f_y (x,\\,y) = – \\frac{1}{y^2} + x\\]
\n\uc774\uba70, \uc774\uacc4\ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[f_{xx}(x,\\,y) = \\frac{2}{x^3} ,\\quad f_{yy} (x,\\,y) = \\frac{2}{y^3} ,\\quad f_{xy} (x,\\,y) = f_{yx} (x,\\,y) = 1\\]
\n\uc774\ub2e4. \uc77c\uacc4\ud3b8\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \\(0\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uc810\uc740 \\((x,\\,y)=(1,\\,1)\\) \ubfd0\uc774\ub2e4. \uc774 \uc810\uc5d0\uc11c Hesse \ud589\ub82c\uc2dd\uc744 \uad6c\ud558\uba74
\n\\[\\det H(f) = 2\\times 2- 1^2 = 3>0\\]
\n\uc774\uace0, \\(f_{xx} (1,\\,1) = 2 > 0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\((1,\\,1)\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n
\n\\[\\lim_{x\\to\\infty} f\\left( x,\\,-x \\right) = \\lim_{x\\to\\infty} \\left( \\frac{1}{x} – x^2 -\\frac{1}{x}\\right) = -\\infty\\]
\n\uc774\ubbc0\ub85c \\(f\\)\ub294 \uc544\ub798\ub85c \uc720\uacc4\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n
\n<\/span>\n<\/p>\n<\/div>\n
\n\\(\\mathbb{R}^2\\)\uc758 \uc81c 1 \uc0ac\ubd84\uba74 \uc601\uc5ed, \uc989 \\(x > 0 ,\\) \\(y > 0\\)\uc778 \uc601\uc5ed\uc5d0\uc11c \ud568\uc218 \\(f\\)\uac00
\n\\[f(x,\\,y) = xy + 2y – \\ln x^2 y\\]
\n\ub85c \uc815\uc758\ub418\uc5c8\uc744 \ub54c, \\(f\\)\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624.
(Thomas’ Calculus 13ed Section 14.7 Problem 42.)\n<\/p>\n<\/div>\n
\n\\(f\\)\uc758 \ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[f_x (x,\\,y) = y+2 – \\frac{2}{x} ,\\quad f_y (x,\\,y) = x – \\frac{1}{y}\\]
\n\uc774\uba70, \uc774\uacc4\ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[f_{xx}(x,\\,y) = \\frac{2}{x^2} ,\\quad f_{yy}(x,\\,y) = \\frac{1}{y^2} ,\\quad f_{xy}(x,\\,y) = f_{yx}(x,\\,y) = 1\\]
\n\uc774\ub2e4. \uc77c\uacc4\ud3b8\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \\(0\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \uc810\uc740
\n\\[(x,\\,y) = \\left( \\frac{1}{2} ,\\, 2 \\right)\\]
\n\ubfd0\uc774\ub2e4. \uc774 \uc810\uc5d0\uc11c Hesse \ud589\ub82c\uc2dd\uc744 \uad6c\ud558\uba74
\n\\[\\det H(f) = 8 \\times \\frac{1}{4} – 1^2 = 1 > 0\\]
\n\uc774\uace0 \\(f_{xx} (\\frac{1}{2} ,\\, 2) = 8 > 0\\)\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(( \\frac{1}{2} ,\\,2)\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n
\n\\[xy – \\ln xy \\ge 1\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\[f(x,\\,y) = 2x – \\ln x + (xy – \\ln xy) \\ge 2x – \\ln x + 1\\]
\n\uc774\ub2e4. \uadf8\ub7f0\ub370
\n\\[\\lim_{x\\to 0+} ( 2x – \\ln x ) = \\infty\\]
\n\uc774\uace0
\n\\[\\lim_{x\\to \\infty} ( 2x – \\ln x ) = \\infty\\]
\n\uc774\ubbc0\ub85c \\((x,\\,y)\\)\uac00 \uc5b4\ub5bb\uac8c \uc6c0\uc9c1\uc774\ub4e0 \\(f\\)\uc758 \uac12\uc774 \\(f(\\frac{1}{2} ,\\,2)\\)\ubcf4\ub2e4 \uc791\uc544\uc9c8 \uc77c\uc740 \uc5c6\ub2e4.<\/p>\n
\n<\/span>\n<\/p>\n<\/div>\n