\\(k\\) \u3092\u5b9a\u6570\u3068\u3059\u308b\u3068\u304d, \u65b9\u7a0b\u5f0f \\(e^x-x^e = k\\) \u306e\u7570\u306a\u308b\u6b63\u306e\u89e3\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/p>\r\n
\\(f(x) = e^x-x^e\\) \u3068\u304a\u3044\u3066, \\(x \\gt 0\\) \u306b\u304a\u3051\u308b\u66f2\u7dda \\(y = f(x)\\) \u3068\u76f4\u7dda \\(y = k\\) \u306e\u4ea4\u70b9\u306e\u500b\u6570\u3092\u8003\u3048\u308c\u3070\u3088\u3044.
\r\n\\(f(x)\\) \u306e\u5c0e\u95a2\u6570\u3092\u9806\u6b21\u6c42\u3081\u308b\u3068\r\n\\[\\begin{align}\r\nf'(x) & = e^x -e x^{e-1} \\\\\r\nf''(x) & = e^x -e(e-1) x^{e-2}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(f(x)\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\left\\{\\begin{array}{l} f(0) = e^0 -0^e = 1 \\\\ f(1) = e^1 -1^e = e-1 \\\\ f(e) = e^e -e^e = 0 \\\\ \\displaystyle\\lim _ {x \\rightarrow \\infty} f(x) = \\infty \\end{array}\\right. \\quad ... [1]\r\n\\]\r\n\u307e\u305f, \\(f'(x)\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\left\\{\\begin{array}{l} f'(0) = e^0 -e \\cdot 0^{e-1} = 1 \\\\ f(1) = e^1 -e \\cdot 1^{e-1} = 0 \\\\ f(e) = e^e -e \\cdot e^{e-1} = 0 \\\\ \\displaystyle\\lim _ {x \\rightarrow \\infty} f'(x) = \\infty \\end{array}\\right. \\quad ... [2]\r\n\\]\r\n[2] \u3088\u308a, \u300c \\(f'(x) = 0\\) \u306f\u5c11\u306a\u304f\u3068\u3082 \\(2\\) \u3064\u306e\u6b63\u306e\u89e3\u3092\u3082\u3064. \u300d ... [3]\r\n\u7d9a\u3044\u3066, \\(f''(x)\\) \u306b\u7740\u76ee\u3057, \\(g(x) = e^x\\) , \\(h(x) = e(e-1) x^{e-2}\\) \u3068\u304a\u304f.
\r\n\\(2 \\lt e \\lt 3\\) \u3067\u3042\u308b\u304b\u3089, \\(y = h(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0a\u306b\u51f8\u3067\u3042\u308b.
\r\n\u4e00\u65b9, \\(y = g(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u306b\u51f8\u306a\u306e\u3067, \\(2\\) \u3064\u306e\u30b0\u30e9\u30d5\u306f\u9ad8\u3005 \\(2\\) \u3064\u306e\u4ea4\u70b9\u3057\u304b\u3082\u305f\u306a\u3044.<\/p>\r\n
1*<\/strong>\u3000\\(2\\) \u3064\u306e\u30b0\u30e9\u30d5\u306e\u4ea4\u70b9\u304c \\(1\\) \u3064\u4ee5\u4e0b\u306e\u3068\u304d 2*<\/strong>\u3000\\(2\\) \u3064\u306e\u30b0\u30e9\u30d5\u306e\u4ea4\u70b9\u304c \\(2\\) \u3064\u306e\u3068\u304d 1*<\/strong> 2*<\/strong>\u3088\u308a, \u300c \\(f'(x) = 0\\) \u306f\u9ad8\u3005 \\(2\\) \u3064\u306e\u6b63\u306e\u89e3\u3057\u304b\u3082\u305f\u306a\u3044. \u300d ... [4]\r\n[3] [4] \u3088\u308a, \\(f'(x) = 0\\) \u306f\u3061\u3087\u3046\u3069 \\(2\\) \u3064\u306e\u6b63\u306e\u89e3\u3092\u3082\u3061, [2]\u3088\u308a, \u305d\u306e\u89e3\u306f\r\n\\[\r\nx = 1 , e\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, [1] \u3068\u3042\u308f\u305b\u3066\u8003\u3048\u308c\u3070, \\(f(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccc} x & (0) & \\cdots & 1 & \\cdots & e & \\cdots & ( \\infty ) \\\\ \\hline f'(x) & & + & 0 & - & 0 & + & \\\\ \\hline f(x) & (1) & \\nearrow & e-1 & \\searrow & 0 & \\nearrow & ( \\infty ) \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u66f2\u7dda \\(y = f(x)\\) \u3068\u76f4\u7dda \\(y = k\\) \u306e\u4f4d\u7f6e\u95a2\u4fc2\u3092\u8003\u3048\u308c\u3070, \u6c42\u3081\u308b\u89e3\u306e\u500b\u6570\u306f\r\n\\[\r\n\\underline{\\left\\{\\begin{array}{ll} 0 & \\left( k \\lt 0 \\text{\u306e\u3068\u304d} \\right) \\\\ 1 & \\left( k = 0 \\text{\u306e\u3068\u304d} \\right) \\\\ 2 & \\left( 0 \\lt k \\leqq 1 \\text{\u306e\u3068\u304d} \\right) \\\\ 3 & \\left( 1 \\lt k \\lt e-1 \\text{\u306e\u3068\u304d} \\right) \\\\ 2 & \\left( k = e-1 \\text{\u306e\u3068\u304d} \\right) \\\\ 1 & \\left( k \\gt e-1 \\text{\u306e\u3068\u304d} \\right) \\end{array}\\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k\\) \u3092\u5b9a\u6570\u3068\u3059\u308b\u3068\u304d, \u65b9\u7a0b\u5f0f \\(e^x-x^e = k\\) \u306e\u7570\u306a\u308b\u6b63\u306e\u89e3\u306e\u500b\u6570\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":[92],"tags":[141,111],"class_list":["post-797","post","type-post","status-publish","format-standard","hentry","category-toko_2013","tag-toko","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/797","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=797"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/797\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=797"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=797"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=797"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\r\n\u5e38\u306b, \\(g(x) \\geqq h(x)\\) \u3059\u306a\u308f\u3061 \\(f''(x) \\geqq 0\\) \u306a\u306e\u3067, \\(f'(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3057\r\n\\[\r\nf'(x) \\gt f'(0) = 1\r\n\\]<\/li>\r\n
\r\n\\(2\\) \u3064\u306e\u4ea4\u70b9\u306e \\(x\\) \u5ea7\u6a19\u3092 \\(\\alpha , \\beta\\) \uff08 \\(\\alpha \\lt \\beta\\) \uff09\u3068\u304a\u304f\u3068, \\(f'(x)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccc} x & (0) & \\cdots & \\alpha & \\cdots & \\beta & \\cdots & ( \\infty ) \\\\ \\hline f''(x) & & + & 0 & - & 0 & + & \\\\ \\hline f'(x) & (1) & \\nearrow & \\text{\u6975\u5927} & \\searrow & \\text{\u6975\u5c0f} & \\nearrow & ( \\infty ) \\end{array}\r\n\\]<\/li>\r\n<\/ol>\r\n