{"id":686,"date":"2023-10-28T16:45:16","date_gmt":"2023-10-28T08:45:16","guid":{"rendered":"http:\/\/ggapa.net:81\/?p=686"},"modified":"2023-11-20T23:10:35","modified_gmt":"2023-11-20T15:10:35","slug":"%e6%b5%85%e8%b0%88%e5%9f%ba%e6%9c%ac%e4%b8%8d%e7%ad%89%e5%bc%8f%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"http:\/\/ggapa.net:81\/2023\/10\/28\/%e6%b5%85%e8%b0%88%e5%9f%ba%e6%9c%ac%e4%b8%8d%e7%ad%89%e5%bc%8f%e9%97%ae%e9%a2%98\/","title":{"rendered":"\u6d45\u8c08\u57fa\u672c\u4e0d\u7b49\u5f0f\u95ee\u9898"},"content":{"rendered":"

$$\\frac{2}{\\frac{1}{a}+\\frac{1}{b}} \\le \\sqrt{ab} \\le \\frac{a+b}{2} \\le \\sqrt{\\frac{a^2+b^2}{2}}$$<\/p>\n

${ \\mathrm{Example\\enspace 1}} $<\/h1>\n

\u95ee\u9898\u63cf\u8ff0<\/h5>\n

\u5df2\u77e5 $x >0, y > 0$ \u4e14 $2x + 8y - xy = 0$ \u6c42 $x + y$ \u7684\u6700\u5c0f\u503c<\/p>\n

\u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n

\u901a\u8fc71\u7684\u4ee3\u6362<\/code>\u6765\u89e3\u51b3\u95ee\u9898<\/p>\n

\u8003\u8651\u5c06 $2x + 8y - xy = 0$ \u5316\u7b80\u53ef\u5f97\uff1a<\/p>\n

$2x + 8y = xy$ \u4e24\u8fb9\u540c\u4e58 $\\frac{1}{xy}$ \u53ef\u5f97\uff1a<\/p>\n

$ \\frac{2}{y} + \\frac{2}{x} =1 $<\/p>\n

$ \\because x+y=(x+y)\\cdot 1 $<\/p>\n

$\\therefore (x+y)=(x+y)(\\frac{8}{x}+\\frac{2}{y} )=\\frac{2x}{y} +\\frac{8y}{x} +10$<\/p>\n

$\\because x>0,y>0 $<\/p>\n

$\\therefore \\frac{2x}{y} +\\frac{8y}{x} \\ge2\\sqrt[]{16} $<\/p>\n

\u5f53\u4e14\u4ec5\u5f53 $x = 6, y = 12$ \u65f6 $(\\frac{2x}{y} +\\frac{8y}{x})_{min}=8$ <\/p>\n

\u6b64\u65f6 $(x+y)_{min}=18$<\/p>\n

${ \\mathrm{Example\\enspace 2}} $<\/h1>\n
\u95ee\u9898\u63cf\u8ff0<\/h5>\n

\u5df2\u77e5 $x\uff0cy> 0,x+2y+xy-6=0$\uff0c\u89e3\u51b3\u4e0b\u5217\u95ee\u9898\uff1a<\/p>\n

    \n
  1. \u6c42 $xy$ \u7684\u6700\u5927\u503c<\/li>\n
  2. $x+2y$\u7684\u6700\u5c0f\u503c<\/li>\n
  3. $x+y$ \u7684\u6700\u5c0f\u503c<\/li>\n
  4. $(x+2)^2+(y+1)^2$\u7684\u6700\u5c0f\u503c<\/li>\n<\/ol>\n
    \u5206\u6790\u4e0e\u89e3\u7b54<\/h5>\n

    $ \\mathrm{{\\Large \u7b2c\u4e00\u95ee} } $<\/p>\n

    \u7b2c\u4e00\u95ee\u53ef\u4ee5\u4f7f\u7528\u57fa\u672c\u4e0d\u7b49\u5f0f\u53ef\u5c06 $x+2y\\Rightarrow \\sqrt{2xy}$ \u7136\u540e\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff1a<\/p>\n

    $\\because x+2y \\ge \\sqrt{2xy},x+2y+xy-6=0$<\/p>\n

    $\\therefore 2\\sqrt{2xy}+xy-6 \\le 0$<\/p>\n

    $\\because x,y>0$<\/p>\n

    $\\therefore (\\sqrt[]{xy} )^2+2\\sqrt[]{2xy} -6\\le 0$<\/p>\n

    $=(\\sqrt[]{xy} )^2+2\\sqrt[]{2} \\sqrt[]{xy} -6\\le 0$<\/p>\n

    \u82e5\u5c06\u4e0a\u9762\u7684\u4e0d\u7b49\u5f0f\u4e2d\u7684 $\\sqrt{xy}$ \u770b\u6210\u4e00\u4e2a\u6574\u4f53\uff0c\u53ef\u5f97\u4e00\u4e2a\u4e00\u5143\u4e8c\u6b21\u4e0d\u7b49\u5f0f\uff0c\u53ef\u89e3\u5f97\uff1a<\/p>\n

    $0 < \\sqrt[]{xy} \\le\\sqrt[]{2} $<\/p>\n

    $0 \\le {xy} \\le 2$<\/p>\n

    $\\therefore xy_{max}=2$<\/p>\n

    \n

    $ \\mathrm{{\\Large \u7b2c\u4e8c\u95ee} } $<\/p>\n

    \u7531\u7b2c\u4e00\u95ee\u53ef\u5f97 $xy_{max}=2$<\/p>\n

    $\\because x+2y=6-xy$<\/p>\n

    $ \\therefore (x+2y)_{max}=6-xy_{min} = 6-2=4$<\/p>\n

    \n

    $ \\mathrm{{\\Large \u7b2c\u4e09\u95ee} } $<\/p>\n

    $\\because x+2y+xy-6=0$<\/p>\n

    $\\therefore 2y+xy=6-x \\Rightarrow y(2+x)=(6-x) \\Rightarrow y=\\frac{6-x}{2+x} $<\/p>\n

    $\\therefore x+y=x+\\frac{6-x}{2+x}=x-\\frac{2+x-8}{2+x} =x-1+\\frac{8}{x+2} =x+2+\\frac{8}{x+2} -3$<\/p>\n

    $\\because x > 0$<\/p>\n

    $\\therefore x+2+\\frac{8}{x+2} \\ge 2\\sqrt[]{8}$ \u5f53\u4e14\u4ec5\u5f53 $x+2=\\frac{8}{x+2}=\\sqrt{8}$ \u65f6\uff0c\u7b49\u53f7\u6210\u7acb\uff0c\u6b64\u65f6 $x=\\sqrt{8}-2$<\/p>\n

    $\\therefore (x+y)_{min}= (x+2+\\frac{8}{x+2})_{min}-3=4\\sqrt[]{2} -3$<\/p>\n

    \n

    $ \\mathrm{{\\Large \u7b2c\u56db\u95ee} } $<\/p>\n

    *\u6b65\u9aa4\u53ef\u80fd\u53ef\u4ee5\u7b80\u5316<\/em><\/p>\n

    \u7531\u7b2c\u4e09\u95ee\u53ef\u77e5 $y=\\frac{6-x}{2+x}, x+2+\\frac{8}{x+2} \\ge 2\\sqrt[]{8}$<\/p>\n

    $\\therefore (x+2)^2+(y+1)=(x+2)^2+(\\frac{6-x}{2+x}+1)=(x+2)^2+(\\frac{8}{2+x})^2=(x+2)^2+\\frac{8^2}{(2+x)^2}$<\/p>\n

    $\\because \\frac{8^2}{(2+x)^2},(x+2)^2\\ge 0$<\/p>\n

    $\\therefore (x+2)^2+\\frac{8^2}{(2+x)^2}\\ge 2\\sqrt[]{8^2} =16$ \u5f53\u4e14\u4ec5\u5f53 $ (x+2)^2=\\frac{8^2}{(2+x)^2}=8$ \u65f6\u6210\u7acb
    \n\u6b64\u65f6 $x=\\sqrt{8}-2$<\/p>\n

    $\\therefore [(x+2)^2+(y+1)^2]_{min}=16$<\/p>\n

    \u603b\u7ed3<\/h5>\n

    \u6b64\u9898\u4e4d\u770b\u53ef\u4ee5\u4f7f1\u7684\u4ee3\u6362<\/code>\u4f46\u6df1\u7a76\u53d1\u73b0\u56e0\u4e3a\u6709\u5e38\u6570\u7684\u5b58\u5728\u800c\u4e0d\u80fd\u4f7f\u7528\u3002\u6b64\u9898\u5229\u7528\u4e86\u6574\u4f53\u601d\u60f3\/\u6362\u5143\u6cd5<\/strong>\u548c\u6d88\u5143\u6cd5<\/strong>\u3002<\/p>\n

      \n
    • \u82e5\u53d1\u73b0\u5206\u6bcd\u8f83\u4e3a\u590d\u6742\u53ef\u4f7f\u7528\u6574\u4f53\u601d\u60f3\/\u6362\u5143\u6cd5<\/strong><\/li>\n
    • \u901a\u8fc7\u56e0\u5f0f\u5206\u89e3<\/strong>\u5c06\u591a\u4e2a\u672a\u77e5\u6570\u5316\u7b80\u4e3a\u4e00\u4e2a\u672a\u77e5\u6570\u6216\u8bb8\u80fd\u65b9\u4fbf\u8fd0\u7b97<\/li>\n
    • \u5206\u5b50\u4e0a\u6709\u672a\u77e5\u6570\u65f6\u4e00\u5b9a\u8981\u8bbe\u6cd5\u5220\u9664<\/li>\n
    • \u6ce8\u610f\u7b26\u53f7\u7684\u6539\u53d8<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"

      $$\\frac{2}{\\frac{1}{a}+\\frac{1}{b}} \\le \\sqrt{ab} \\le \\ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[26],"_links":{"self":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/686"}],"collection":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/comments?post=686"}],"version-history":[{"count":27,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/686\/revisions"}],"predecessor-version":[{"id":973,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/posts\/686\/revisions\/973"}],"wp:attachment":[{"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/media?parent=686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/categories?post=686"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ggapa.net:81\/wp-json\/wp\/v2\/tags?post=686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}