\ub2e4\uc74c\uacfc \uac19\uc740 \ud0c0\uc6d0\uc758 \ubc29\uc815\uc2dd\uc744 \uc0dd\uac01\ud574 \ubd05\uc2dc\ub2e4.
\n\\[2x^2 – 4xy + 5y^2 = 36\\tag{1}\\]
\n\uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c \uacf5\ubd80\ud55c \uc774\ucc28\ud615\uc2dd\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74 \uc88c\ubcc0\uc744 \ubcc0\ud615\ud558\uc5ec \ud0c0\uc6d0\uc758 \uc7a5\ucd95\uacfc \ub2e8\ucd95\uc758 \uae38\uc774\ub97c \uad6c\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \uc624\ub298\uc740 \ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \ud0c0\uc6d0\uc758 \uc7a5\ucd95\uacfc \ub2e8\ucd95\uc758 \uae38\uc774\ub97c \uad6c\ud574\ubcf4\uaca0\uc2b5\ub2c8\ub2e4.<\/p>\n
\ud0c0\uc6d0\uc758 \uc911\uc2ec\uc774 \uc88c\ud45c\ud3c9\uba74\uc758 \uc6d0\uc810\uc774\ubbc0\ub85c, \ud0c0\uc6d0 \uc704\uc758 \uc810 \uc911\uc5d0\uc11c \uc6d0\uc810\uc73c\ub85c\ubd80\ud130 \uac00\uc7a5 \uba40\ub9ac \uc788\ub294 \uc810\uae4c\uc9c0\uc758 \uac70\ub9ac\uc640 \uac00\uc7a5 \uac00\uae4c\uc774 \uc788\ub294 \uc810\uae4c\uc9c0\uc758 \uac70\ub9ac\ub97c \ucc3e\uc73c\uba74 \ub429\ub2c8\ub2e4. \uc989 \ud0c0\uc6d0 \uc704\uc758 \uc810 \\((x,\\,y)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \ud568\uc218
\n\\[f(x,\\,y) = x^2 + y^2\\tag{2}\\]
\n\uc758 \uac12\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \ucc3e\uc73c\uba74 \ub429\ub2c8\ub2e4. (\ubb3c\ub860 \uc6d0\uc810\uc73c\ub85c\ubd80\ud130 \uadf8 \uc810\uae4c\uc9c0 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub294 \uc704 \ud568\uc22b\uac12\uc758 \uc81c\uacf1\uadfc\uacfc \uac19\uc2b5\ub2c8\ub2e4.) \ud0c0\uc6d0\uc758 \ubc29\uc815\uc2dd\uc758 \uc88c\ubcc0\uc744 \\(g(x,\\,y)\\)\ub85c \ub450\uace0 \ub2e4\uc74c\uacfc \uc81c\ud55c\uc870\uac74\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0c5\uc2dc\ub2e4.
\n\\[g(x,\\,y) = 36\\tag{3}\\]
\n\uc774\uc81c \ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\uae30 \uc704\ud574 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\uc815\uc2dd\uc744 \uc0dd\uac01\ud569\ub2c8\ub2e4.
\n\\[\\lambda \\nabla f = \\nabla g.\\tag{4}\\]
\n\uc5ec\uae30\uc11c \\(\\lambda\\)\ub294 \uc2e4\uc218\uc785\ub2c8\ub2e4. [\ubcf4\ud1b5\uc740 \\(\\lambda\\)\ub97c \\(\\nabla g\\) \uc55e\uc5d0 \ubd99\uc774\uc9c0\ub9cc \uc5ec\uae30\uc11c\ub294 \uace0\uc733\uac12\uacfc \uace0\uc720\ubca1\ud130\ub97c \uc774\uc6a9\ud558\uae30 \uc704\ud558\uc5ec \\(f\\) \uc55e\uc5d0 \ubd99\uc600\uc2b5\ub2c8\ub2e4.] \uc774 \uc2dd\uc744 \ud480\uc5b4 \uc4f0\uba74
\n\\[\\lambda (2x\\mathbf{i} + 2y\\mathbf{j}) = (4x-4y)\\mathbf{i} + (-4x+10y)\\mathbf{j}\\]
\n\uc989
\n\\[\\lambda (x\\mathbf{i} + y\\mathbf{j}) = (2x-2y)\\mathbf{i} + (-2x+5y)\\mathbf{j}\\]
\n\uc785\ub2c8\ub2e4. \uc774 \uc2dd\uc744 \ud589\ub82c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.
\n\\[\\lambda\\left[\\begin{array}{c} x\\\\y \\end{array}\\right]
\n= \\left[\\begin{array}{rr} 2 & -2 \\\\ -2 & 5 \\end{array}\\right]
\n\\left[\\begin{array}{c} x \\\\ y \\end{array}\\right]\\tag{5}\\]
\n\uc6b0\ubcc0\uc758 \uc815\uc0ac\uac01\ud589\ub82c\uc744 \\(A\\)\ub85c \ub450\uba74, \uc704 \uc2dd\uc740 \ud589\ub82c \\(A\\)\uc758 \uace0\uc733\uac12 \\(\\lambda\\)\uc640 \uace0\uc720\ubca1\ud130 \\((x,\\,y)\\)\ub97c \uad6c\ud558\ub294 \uc2dd\uc785\ub2c8\ub2e4. \\(A\\)\uc758 \uace0\uc733\uac12\uc744 \uad6c\ud558\uae30 \uc704\ud558\uc5ec \\(A\\)\uc758 \ud2b9\uc131\ub2e4\ud56d\uc2dd \\(p(t)\\)\ub97c \uc774\uc6a9\ud569\uc2dc\ub2e4.
\n\\[\\begin{aligned}
\np(t)
\n&= \\det\\left[\\begin{array}{cc} t-2 & 2 \\\\ 2 & t-5 \\end{array}\\right] \\\\[4pt]
\n&= (t-2)(t-6) -4 \\\\[6pt]
\n&= t^2 – 7t +6 \\\\[6pt]
\n&= (t-1)(t-6) =0.
\n\\end{aligned}\\]
\n\uc774 \ubc29\uc815\uc2dd\uc744 \ud480\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \uace0\uc733\uac12\uc744 \uc5bb\uc2b5\ub2c8\ub2e4.
\n\\[\\lambda_1 = 1 ,\\,\\, \\lambda_2 = 6.\\tag{6}\\]
\n\uc774 \uace0\uc733\uac12\uc5d0 \ub300\uc751\ub418\ub294 \uace0\uc720\ubca1\ud130\ub294 \ucc28\ub840\ub85c
\n\\[v_1 = k_1 \\left[\\begin{array}{c} 2\\\\1 \\end{array}\\right] ,\\,\\,
\nv_2 = k_2 \\left[\\begin{array}{r} -1\\\\2 \\end{array}\\right]\\tag{7}\\]
\n\uc785\ub2c8\ub2e4.<\/p>\n
\uba3c\uc800 \\(v_1\\)\uc744 \uc81c\ud55c\uc870\uac74 (3)\uc5d0 \ub300\uc785\ud558\uba74
\n\\[g(v_1 ) = 8k_1 ^2 – 8k_1 ^2 + 5k_1 ^2 = 36\\]
\n\uc774\ubbc0\ub85c
\n\\[k_1 = \\pm\\frac{6}{\\sqrt{5}}\\]
\n\ub97c \uc5bb\uc2b5\ub2c8\ub2e4. \uc774\ub807\uac8c \uc5bb\uc740 \\(k_1\\)\uc5d0 \ub300\ud558\uc5ec \\(v_1\\)\uc744 \\(f\\)\uc5d0 \ub300\uc785\ud558\uba74
\n\\[f(v_1) = \\left(\\frac{12}{\\sqrt{5}}\\right)^2 + \\left(\\frac{6}{\\sqrt{5}}\\right)^2 = 36\\]
\n\uc785\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud0c0\uc6d0\uc758 \ub450 \ucd95 \uc911 \ud55c \ucd95\uc758 \uae38\uc774\ub294
\n\\[2 \\times \\sqrt{f(v_1)} = 2 \\times 6 = 12\\tag{8}\\]
\n\uc785\ub2c8\ub2e4.<\/p>\n
\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(v_2\\)\ub97c \uc81c\ud55c\uc870\uac74 (3)\uc5d0 \ub300\uc785\ud558\uba74
\n\\[g(v_2) = 30k_2 ^2 = 36\\]
\n\uc774\ubbc0\ub85c
\n\\[k_2 = \\pm\\sqrt{\\frac{6}{5}}\\]
\n\ub97c \uc5bb\uc2b5\ub2c8\ub2e4. \uc774\ub807\uac8c \uc5bb\uc740 \\(k_2\\)\uc5d0 \ub300\ud558\uc5ec \\(v_2\\)\ub97c \\(f\\)\uc5d0 \ub300\uc785\ud558\uba74
\n\\[f(v_2)= k_2 ^2 + 4k_2 ^2 = 6\\]
\n\uc785\ub2c8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud0c0\uc6d0\uc758 \ub450 \ucd95 \uc911 \ub2e4\ub978 \ud55c \ucd95\uc758 \uae38\uc774\ub294
\n\\[2\\times \\sqrt{f(v_2)} = 2 \\times \\sqrt{6} = 2\\sqrt{6}\\tag{9}\\]
\n\uc785\ub2c8\ub2e4.<\/p>\n
(8)\uacfc (9)\ub97c \ube44\uad50\ud558\uba74 (8)\uc5d0\uc11c \uad6c\ud55c \uac12\uc774 \ub354 \ud06c\ubbc0\ub85c, \uc8fc\uc5b4\uc9c4 \ud0c0\uc6d0\uc758 \uc7a5\ucd95\uc758 \uae38\uc774\ub294 \\(12\\)\uc774\uace0 \ub2e8\ucd95\uc758 \uae38\uc774\ub294 \\(2\\sqrt{6}\\)\uc785\ub2c8\ub2e4.<\/p>\n
\ubb3c\ub860 \uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c \uc0b4\ud3b4\ubcf4\uba74 \ubc29\ubc95\uc744 \ud65c\uc6a9\ud558\uba74 (1)\uc740 \uc801\uc808\ud788 \uc88c\ud45c\ub97c \ubcc0\ud615\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ubc14\uafc0 \uc218 \uc788\uc2b5\ub2c8\ub2e4.
\n\\[1(x’)^2 + 6(y’)^2 = 36\\]
\n\uc5ec\uae30\uc11c \uc774\ucc28\ud56d\uc758 \uacc4\uc218 \\(1\\)\uacfc \\(6\\)\uc740 (6)\uc5d0\uc11c \uad6c\ud55c \ub450 \uace0\uc733\uac12\uc785\ub2c8\ub2e4. \uc774 \uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \ud0c0\uc6d0\uc758 \ub450 \ucd95\uc758 \uae38\uc774\ub97c \ubc14\ub85c \uad6c\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\ub2e4\uc74c\uacfc \uac19\uc740 \ud0c0\uc6d0\uc758 \ubc29\uc815\uc2dd\uc744 \uc0dd\uac01\ud574 \ubd05\uc2dc\ub2e4. \\(2x^2 – 4xy + 5y^2 = 36\\) \uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c \uacf5\ubd80\ud55c \uc774\ucc28\ud615\uc2dd\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uba74 \uc88c\ubcc0\uc744 \ubcc0\ud615\ud558\uc5ec \ud0c0\uc6d0\uc758 \uc7a5\ucd95\uacfc \ub2e8\ucd95\uc758 \uae38\uc774\ub97c \uad6c\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ud558\uc9c0\ub9cc \uc624\ub298\uc740 \ub77c\uadf8\ub791\uc8fc\uc758 \ubc29\ubc95(method of Lagrange’s multiplier)\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \ud0c0\uc6d0\uc758 \uc7a5\ucd95\uacfc \ub2e8\ucd95\uc758 \uae38\uc774\ub97c \uad6c\ud574\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. \ud0c0\uc6d0\uc758 \uc911\uc2ec\uc774 \uc88c\ud45c\ud3c9\uba74\uc758 \uc6d0\uc810\uc774\ubbc0\ub85c, \ud0c0\uc6d0 \uc704\uc758 \uc810 \uc911\uc5d0\uc11c \uc6d0\uc810\uc73c\ub85c\ubd80\ud130 \uac00\uc7a5 \uba40\ub9ac \uc788\ub294 \uc810\uae4c\uc9c0\uc758 \uac70\ub9ac\uc640 \uac00\uc7a5 \uac00\uae4c\uc774 \uc788\ub294 \uc810\uae4c\uc9c0\uc758 \uac70\ub9ac\ub97c \ucc3e\uc73c\uba74 \ub429\ub2c8\ub2e4. \uc989 \ud0c0\uc6d0…<\/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":[47,57],"tags":[387],"class_list":["post-6580","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","category-linear-algebra","tag-lagrangian-multiplier"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6580","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=6580"}],"version-history":[{"count":17,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6580\/revisions"}],"predecessor-version":[{"id":6597,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/6580\/revisions\/6597"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6580"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=6580"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=6580"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}