\\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc774\uace0 \\(x_1 ,\\) \\(x_2 ,\\) \\(\\cdots ,\\) \\(x_n\\)\uc774 \ubaa8\ub450 \\(0\\) \uc774\uc0c1\uc778 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\sqrt[n]{x_1 x_2 \\cdots x_n} \\le \\frac{x_1 + x_2 + \\cdots + x_n}{n}\\tag{1}\\]
\n\uc5ec\uae30\uc11c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x_1 = x_2 = \\cdots = x_n\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n
\ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774\uac83\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc99d\uba85\uc744 \ub9c8\uce60 \ub54c\uae4c\uc9c0 \ucca8\uc218 \\(i\\)\ub294 \\(n\\) \uc774\ud558\uc758 \uc790\uc5f0\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.<\/p>\n
\ub9cc\uc57d \\(x_i\\) \uc911\uc5d0\uc11c \\(0\\)\uc778 \uac83\uc774 \uc788\uc73c\uba74 \uc99d\uba85\uc740 \uc790\uba85\ud558\uac8c \ub05d\ub09c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubaa8\ub4e0 \\(x_i\\)\uac00 \uc591\uc218\ub77c\uace0 \uac00\uc815\ud558\uc790.
\n\\[x = \\sqrt[n]{x_1 x_2 \\cdots x_n}\\tag{2}\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(x > 0\\)\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c
\n\\[y_i = \\frac{x_i}{x}\\tag{3}\\]
\n\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[y_1 y_2 \\cdots y_n = 1\\tag{4}\\]
\n\uc774\ub2e4. \uc774\uc81c \uc591\uc218 \\(y_i\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(F\\)\uc640 \\(G\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[\\begin{gather}
\nF( y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n ) = \\frac{y_1 + y_2 + \\cdots + y_n}{n},\\\\[8pt]
\nG( y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n ) = y_1 y_2 \\cdots y_n .
\n\\end{gather}\\]
\n\uadf8\ub9ac\uace0 \uc9d1\ud569 \\(K_1 ,\\) \\(K_2 \\)\ub97c \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
\n\\[\\begin{gather}
\nK_1 = \\left\\{ (y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n ) \\,\\vert\\, \\lvert y_1 \\rvert \\le n ,\\, \\lvert y_2 \\rvert \\le n ,\\,\\cdots,\\, \\lvert y_n\\rvert \\le n \\right\\}, \\\\[8pt]
\nK_2 = \\left\\{ (y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n ) \\,\\vert\\, G(y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n ) = 1 \\right\\}.
\n\\end{gather}\\]
\n\uc774\uc81c \uc6b0\ub9ac\ub294 \uc81c\uc57d\uc870\uac74
\n\\[G(y_1 ,\\, y_2 ,\\, \\cdots ,\\,y_n ) = 1\\]
\n\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud560 \uac83\uc774\ub2e4. \uc774\uac83\uc740 \uc9d1\ud569 \\(K_2\\)\uc5d0\uc11c \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \uac83\uacfc \uac19\ub2e4. \uadf8\ub7f0\ub370 \\(y_i\\) \uc911\uc5d0\uc11c \\(n\\)\ubcf4\ub2e4 \ud070 \uac83\uc774 \ud558\ub098\ub77c\ub3c4 \uc874\uc7ac\ud55c\ub2e4\uba74 \\(F\\)\uc758 \uac12\uc740 \\(1\\)\ubcf4\ub2e4 \ucee4\uc9c4\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \\(y_i\\)\uc758 \uac12\uc774 \\(1\\)\uc77c \ub54c
\n\\[F(1,\\,1,\\,\\cdots,\\,1) = 1\\]
\n\uc774\ubbc0\ub85c \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc740 \\(1\\) \uc774\ud558\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(K_2\\)\uc5d0\uc11c \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \uac83\uc740 \uacb0\uad6d \\(K_1 \\cap K_2\\)\uc5d0\uc11c \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \uac83\uacfc \uac19\ub2e4. \ud2b9\ud788 \\(K_1 \\cap K_2\\)\ub294 \uc720\uacc4\uc774\uace0 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(F\\)\ub294 \\(K_1 \\cap K_2\\)\uc5d0\uc11c \ubc18\ub4dc\uc2dc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n
\uc774\uc81c \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95\uc744 \uc774\uc6a9\ud558\uc790. \\(F\\)\uc758 \ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ubb38\uc81c. \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \\(K_2\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud574 \ubcf4\uc138\uc694.<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" \\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc774\uace0 \\(x_1 ,\\) \\(x_2 ,\\) \\(\\cdots ,\\) \\(x_n\\)\uc774 \ubaa8\ub450 \\(0\\) \uc774\uc0c1\uc778 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\sqrt[n]{x_1 x_2 \\cdots x_n} \\le \\frac{x_1 + x_2 + \\cdots + x_n}{n}\\) \uc5ec\uae30\uc11c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x_1 = x_2 = \\cdots = x_n\\)\uc778 \uac83\uc774\ub2e4. \ub77c\uadf8\ub791\uc8fc \uc2b9\uc218\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774\uac83\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc99d\uba85\uc744 \ub9c8\uce60 \ub54c\uae4c\uc9c0 \ucca8\uc218 \\(i\\)\ub294 \\(n\\) \uc774\ud558\uc758 \uc790\uc5f0\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud55c\ub2e4.…<\/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":[386,398,396,399,397],"class_list":["post-4435","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-lagrangian-method","tag-398","tag-396","tag-399","tag-397"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4435","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=4435"}],"version-history":[{"count":17,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4435\/revisions"}],"predecessor-version":[{"id":4541,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4435\/revisions\/4541"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4435"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4435"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4435"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\[\\frac{\\partial F}{\\partial y_i} = \\frac{1}{n}\\]
\n\uc774\uba70, \\(G\\)\uc758 \ud3b8\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uace0 \uc81c\uc57d\uc870\uac74\uc744 \uc774\uc6a9\ud558\uc5ec \ubcc0\ud615\ud558\uba74
\n\\[\\frac{\\partial G}{\\partial y_i} = \\frac{G(y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n )}{y_i} = \\frac{1}{y_i}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F\\)\uac00 \ucd5c\uc19f\uac12\uc744 \uac16\ub294 \uc810\uc5d0\uc11c, \uc0c1\uc218 \\(\\lambda\\)\uac00 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\frac{1}{n} = \\lambda \\frac{1}{y_i},\\]
\n\uc989 \\(y_i = n\\lambda\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7f0\ub370 \\(n\\lambda\\)\ub294 \uc0c1\uc218\uc774\ubbc0\ub85c, \ubaa8\ub4e0 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(y_i\\)\ub294 \uac19\uc740 \uac12\uc774\ub2e4. \ub530\ub77c\uc11c \\(F\\)\ub294
\n\\[y_1 = y_2 = \\cdots = y_n \\]
\n\uc778 \uc810 \\((y_1 ,\\, y_2 ,\\, \\cdots ,\\, y_n )\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \ub9cc\uc57d
\n\\[y_1 = y_2 = \\cdots = y_n = 1\\]
\n\uc774\uba74 \\(F(y_1 ,\\,y_2 ,\\, \\cdots ,\\, y_n ) = 1\\)\uc774\ubbc0\ub85c \\(1\\)\uc740 \\(F\\)\uc758 \ucd5c\uc19f\uac12\uc774\ub2e4.
\n\ub05d\uc73c\ub85c
\n\\[\\begin{align}
\n\\sqrt[n]{x_1 x_2 \\cdots x_n}
\n= x
\n&\\le x \\frac{y_1 + y_2 + \\cdots + y_n}{n} \\\\[6pt]
\n&= \\frac{xy_1 + xy_2 + \\cdots + xy_n}{n} \\\\[6pt]
\n&= \\frac{x_1 + x_2 + \\cdots + x_n}{n}
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c \uc6d0\ud558\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n