\u5b9f\u6570 \\(x , y\\) \u304c\u6761\u4ef6 \\(x^2 +xy +y^2 =6\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u304d\r\n\\[\r\nx^2y +xy^2 -x^2 -2xy -y^2 +x +y\r\n\\]\r\n\u304c\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(P =x^2y +xy^2 -x^2 -2xy -y^2 +x +y\\) \u3068\u304a\u304f.
\r\n\\(s=x+y\\) , \\(t=xy\\) \u3068\u304a\u304f.
\r\n\\(x , y\\) \u306f\u65b9\u7a0b\u5f0f \\(u^2 -su +t =0\\) \u306e \\(2\\) \u3064\u306e\u5b9f\u6570\u89e3\u306a\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nD =s^2 -4t \\geqq 0 \\quad ... [1]\r\n\\]\r\n\u307e\u305f, \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\nx^2 +xy +y^2 & = s^2 -t = 6 \\\\\r\n\\text{\u2234} \\quad t & =s^2 -6 \\quad ... [2]\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\ns^2 -4 \\left( s^2 -6 \\right) & \\geqq 0 \\\\\r\ns^2 -8 & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad -2 \\sqrt{2} \\leqq s & \\leqq 2 \\sqrt{2} \\quad ... [3]\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nP & = st -s^2 +s =s \\left( s^2 -6 \\right) -s^2 +s \\\\\r\n& = s^3 -s^2 -5s\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 \\(f(s)\\) \u3068\u304a\u3044\u3066\r\n\\[\r\nf'(s) =3s^2 -2s -5 =(3s-5)(s+1)\r\n\\]\r\n\\(f'(s)=0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\ns = -1 , \\dfrac{5}{3}\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nf \\left( -2 \\sqrt{2} \\right) & = -16 \\sqrt{2} -8 +10 \\sqrt{2} \\\\\r\n& = -8 -6 \\sqrt{2} , \\\\\r\nf (-1) & = -1 -1 +5 =3 , \\\\\r\nf \\left( \\dfrac{5}{3} \\right) & = \\dfrac{125}{27} -\\dfrac{25}{9} -\\dfrac{25}{3} =-\\dfrac{175}{27} , \\\\\r\nf \\left( 2 \\sqrt{2} \\right) & = 16 \\sqrt{2} -8 -10 \\sqrt{2} \\\\\r\n& = -8 +6 \\sqrt{2}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u7bc4\u56f2 [3] \u306b\u304a\u3051\u308b \\(f(s)\\) \u306e\u5897\u6e1b\u306f, \u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccccc} s & -2 \\sqrt{2} & \\cdots & -1 & \\cdots & \\dfrac{5}{3} & \\cdots & 2 \\sqrt{2} \\\\ \\hline f'(s) & & + & 0 & - & 0 & + & \\\\ \\hline f(s) & -8 -6 \\sqrt{2} & \\nearrow & 3 & \\searrow & -\\dfrac{175}{27} & \\nearrow & -8 +6 \\sqrt{2} \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf \\left( \\dfrac{5}{3} \\right) & \\gt -6 \\gt f \\left( -2 \\sqrt{2} \\right) , \\\\\r\nf \\left( 2 \\sqrt{2} \\right) & \\lt -8 +6 \\cdot 1.5 =1 \\lt f(-1)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u3088\u3063\u3066 \\(P\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{-8 -6 \\sqrt{2} \\leqq P \\leqq 3}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(x , y\\) \u304c\u6761\u4ef6 \\(x^2 +xy +y^2 =6\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089\u52d5\u304f\u3068\u304d \\[ x^2y +xy^2 -x^2 -2xy -y^2 +x +y \\] \u304c\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[59],"tags":[140,68],"class_list":["post-371","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2012","tag-kyoto_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/371","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=371"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/371\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=371"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=371"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=371"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}