{"id":499,"date":"2026-06-10T15:39:28","date_gmt":"2026-06-10T07:39:28","guid":{"rendered":"https:\/\/blog.hcha.site\/?p=499"},"modified":"2026-06-10T18:27:12","modified_gmt":"2026-06-10T10:27:12","slug":"%e5%87%ba%e9%a2%98%e7%9a%84%e7%a6%bb%e7%a5%9e%e4%b8%8d%e8%bf%9c%e4%ba%86-2026-%e6%96%b0%e9%ab%98%e8%80%83%e2%85%b0%e5%8d%b7-%c2%b7-%e7%ac%ac19%e9%a2%98%e8%a7%a3%e6%9e%90","status":"publish","type":"post","link":"https:\/\/blog.hcha.site\/?p=499","title":{"rendered":"\u51fa\u9898\u7684\u79bb\u795e\u4e0d\u8fdc\u4e86 \u2014\u2014 2026 \u65b0\u9ad8\u8003\u2160\u5377 \u00b7 \u7b2c19\u9898\u89e3\u6790"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>19. (17\u5206)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5df2\u77e5\u51fd\u6570 $f(x)$ \u7684\u5b9a\u4e49\u57df\u4e3a $\\mathbf{R}$\uff0c\u4e14\u5f53 $x&lt;0$ \u65f6\uff0c$f(x) = 2^x$\u3002\u5bf9\u4efb\u610f $x_0 \\in \\mathbf{R}$\uff0c\u5b9a\u4e49\u96c6\u5408 $D(x_0) = \\{d \\in \\mathbf{R} \\mid f(x_0+d) &gt; f(x_0)\\}$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff081\uff09\u82e5\u5f53 $x \\geqslant 0$ \u65f6\uff0c$f(x) = 1-x$\uff0c\u6c42 $D(-1)$\uff1b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff082\uff09\u82e5 $f(x)$ \u662f\u5947\u51fd\u6570\uff0c$f(x_1) \\leqslant f(x_2)$\uff0c\u4e14 $x_1 x_2 \\neq 0$\uff0c\u8bc1\u660e\uff1a$D(x_2) \\subseteq D(x_1)$\uff1b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff083\uff09\u8bbe $f(x)$ \u6ee1\u8db3\uff1a\u2460 \u82e5 $f(x_1) \\leqslant f(x_2)$\uff0c\u5219 $D(x_2) \\subseteq D(x_1)$\uff1b\u2461 \u5f53 $0&lt;x&lt;1$ \u65f6\uff0c$f(x) &lt; f(0)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff08i\uff09\u8bc1\u660e\uff1a$f(0) \\geqslant 1$\uff1b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\uff08ii\uff09\u8bc1\u660e\uff1a$f(x)$ \u5728\u533a\u95f4 $(0, +\\infty)$ \u5355\u8c03\u9012\u589e\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">0. D \u662f\u4ec0\u4e48\u9b3c\u73a9\u610f<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">$$D(x_0) = \\{ d \\in \\mathbb{R} \\mid f(x_0 + d) &gt; f(x_0) \\}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7ffb\u8bd1\u6210\u4e2d\u6587\uff1a<strong>\u4ece $x_0$ \u51fa\u53d1\uff0c\u5f80\u5de6\u6216\u5f80\u53f3\u504f\u79fb $d$\uff0c\u5982\u679c\u51fd\u6570\u503c\u53d8\u5927\u4e86\uff0c\u90a3\u8fd9\u4e2a $d$ \u5c31\u5c5e\u4e8e $D(x_0)$\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $D(x_0)$ \u6536\u96c6\u7684\u662f\u6240\u6709\u80fd\u8ba9\u51fd\u6570\u503c&#8221;\u53d8\u5f97\u66f4\u5927&#8221;\u7684\u504f\u79fb\u91cf\u3002\u76f4\u89c9\u4e0a\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c $f$ \u5728 $x_0$ \u9644\u8fd1\u662f\u9012\u589e\u7684\uff0c\u90a3\u6b63\u7684\u5c0f $d$ \u5c31\u5728 $D(x_0)$ \u91cc\u3002<\/li>\n\n\n\n<li>\u5982\u679c $f(x_0)$ \u7684\u503c\u5df2\u7ecf\u5f88\u5927\u4e86\uff0c\u80fd\u8ba9\u5b83\u66f4\u5927\u7684 $d$ \u5c31\u5f88\u5c11\uff0c$D(x_0)$ \u5c31\u5c0f\u3002<\/li>\n\n\n\n<li>\u5982\u679c $f(x_0)$ \u7684\u503c\u5f88\u5c0f\uff0c\u968f\u4fbf\u4e00\u4e2a\u504f\u79fb\u90fd\u53ef\u80fd\u8ba9\u51fd\u6570\u503c\u66f4\u5927\uff0c$D(x_0)$ \u5c31\u5927\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u8fd9\u4e2a\u76f4\u89c9\u6b63\u662f\u7b2c(2)(3)\u9898\u4e2d&#8221;$f(x_1) \\leq f(x_2) \\Rightarrow D(x_2) \\subseteq D(x_1)$&#8221;\u7684\u542b\u4e49\uff1a<strong>\u51fd\u6570\u503c\u8d8a\u5927\u7684\u5730\u65b9\uff0c&#8221;\u8fd8\u80fd\u53d8\u5f97\u66f4\u5927&#8221;\u7684\u4f59\u5730\u8d8a\u5c0f\u3002<\/strong><br><br><strong><strong>\u4f60\u7ad9\u5f97\u8d8a\u9ad8\uff0c\u8fd8\u80fd\u5f80\u4e0a\u8d70\u7684\u8def\u5c31\u8d8a\u5c11\u3002\u4f60\u7ad9\u5f97\u8d8a\u4f4e\uff0c\u80fd\u5f80\u4e0a\u8d70\u7684\u8def\u5c31\u8d8a\u591a\u3002<\/strong><\/strong>\u505a\u9053\u9ad8\u8003\u9898\u8fd8\u80fd\u505a\u51fa\u4eba\u751f\u54f2\u7406\u51fa\u6765\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53e6\u5916\uff0c<strong>\u5f53 $x &lt; 0$ \u65f6\uff0c$f(x) = 2^x$\u3002<\/strong> \u4e5f\u5c31\u662f\u8bf4\u8d1f\u534a\u8f74\u4e0a\u7684\u884c\u4e3a\u662f\u5b8c\u5168\u786e\u5b9a\u7684\u2014\u2014\u4e00\u4e2a\u503c\u57df\u4e3a $(0,1)$ \u7684\u4e25\u683c\u9012\u589e\u51fd\u6570\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u7b2c(1)\u95ee\uff1a\u9001\u5206\u9898<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">1.1 \u660e\u786e\u76ee\u6807<\/h3>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mo fence=\"true\" form=\"prefix\">{<\/mo><mtable><mtr><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><msup><mn>2<\/mn><mi>x<\/mi><\/msup><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>x<\/mi><mo>\u2265<\/mo><mn>0.<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><mo fence=\"true\" form=\"postfix\"><\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = \\begin{cases} \n2^x, &amp; x &lt; 0, \\\\ \n1 &#8211; x, &amp; x \\geq 0. \n\\end{cases}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u56e0\u4e3a $f(-1) = 2^{-1} = \\dfrac{1}{2}$\uff0c\u6240\u4ee5\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$d \\in D(-1) \\iff f(-1 + d) &gt; \\frac{1}{2}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee4 $t = -1 + d$\uff08\u5373 $d = t + 1$\uff09\uff0c\u95ee\u9898\u53d8\u6210\uff1a<strong>\u6c42\u6240\u6709\u4f7f $f(t) &gt; \\dfrac{1}{2}$ \u7684 $t$\uff0c\u7136\u540e\u5e73\u79fb\u56de $d$\u3002<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.2 \u5206\u6bb5\u6c42\u89e3<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5f53 $t &lt; 0$ \u65f6<\/strong>\uff08\u5bf9\u5e94 $d &lt; 1$\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(t) = 2^t &gt; \\frac{1}{2} = 2^{-1} \\iff t &gt; -1$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $-1 &lt; t &lt; 0$\uff0c\u5373 $0 &lt; d &lt; 1$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5f53 $t \\geq 0$ \u65f6<\/strong>\uff08\u5bf9\u5e94 $d \\geq 1$\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(t) = 1 &#8211; t &gt; \\frac{1}{2} \\iff t &lt; \\frac{1}{2}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $0 \\leq t &lt; \\dfrac{1}{2}$\uff0c\u5373 $1 \\leq d &lt; \\dfrac{3}{2}$\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.3 \u5408\u5e76<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$$D(-1) = (0, 1) \\cup \\left[1, \\frac{3}{2}\\right) = \\boxed{\\left(0,\\;\\frac{3}{2}\\right)}$$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">\u7b2c(1)\u95ee\u672c\u8d28\u4e0a\u5c31\u662f\u89e3\u4e0d\u7b49\u5f0f $f(t) &gt; \\dfrac{1}{2}$\u3002\u552f\u4e00\u8981\u6ce8\u610f\u7684\u662f\u5206\u6bb5\u51fd\u6570\u7684<strong>\u5206\u6bb5\u8ba8\u8bba<\/strong>\u548c\u6700\u540e\u7684<strong>\u533a\u95f4\u5408\u5e76<\/strong>\u3002<\/p>\n<\/blockquote>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u7b2c(2)\u95ee\uff1a\u521d\u89c1\u7aef\u502a<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f53\u4f60\u53d1\u73b0\u8fd9\u9898\u51fa\u73b0\u5728\u7b2c\u4e8c\u95ee\u7684\u65f6\u5019\u5c31\u5e94\u8be5\u610f\u8bc6\u5230\u4e0d\u5bf9\uff0c\u53ef\u60dc\u6211\u6ca1\u6709\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.1 \u5148\u628a\u5b8c\u6574\u8868\u8fbe\u5f0f\u5199\u51fa\u6765<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">$f$ \u662f\u5947\u51fd\u6570\uff0c\u6240\u4ee5 $f(0) = 0$\uff0c\u4e14\u5f53 $x &gt; 0$ \u65f6\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(x) = -f(-x) = -2^{-x}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5c55\u5f00\uff1a<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mo fence=\"true\" form=\"prefix\">{<\/mo><mtable><mtr><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><msup><mn>2<\/mn><mi>x<\/mi><\/msup><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd class=\"tml-left\" style=\"padding-left:0em;padding-right:0em;\"><mrow><mo>\u2212<\/mo><msup><mn>2<\/mn><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>x<\/mi><\/mrow><\/msup><mo separator=\"true\">,<\/mo><\/mrow><\/mtd><mtd class=\"tml-left\" style=\"padding-left:1em;padding-right:0em;\"><mrow><mi>x<\/mi><mo>&gt;<\/mo><mn>0.<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><mo fence=\"true\" form=\"postfix\"><\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = \\begin{cases} \n2^x, &amp; x &lt; 0, \\\\ \n0, &amp; x = 0, \\\\ \n-2^{-x}, &amp; x &gt; 0. \n\\end{cases}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.2 \u5206\u522b\u7b97\u51faD(x)\u7684\u8868\u8fbe\u5f0f<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3a\u4e86\u8bc1\u660e\u96c6\u5408\u5305\u542b\u5173\u7cfb\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053 $D(x)$ \u957f\u4ec0\u4e48\u6837\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u60c5\u5f62\u4e00\uff1a$x &lt; 0$\u3002<\/strong> \u6b64\u65f6 $f(x) = 2^x \\in (0,1)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5bf9 $d$ \u7684\u504f\u79fb $t = x + d$\uff0c\u5206\u4e09\u6bb5\u8ba8\u8bba $f(t) &gt; 2^x$\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u82e5 $t &lt; 0$\uff1a$2^t &gt; 2^x \\iff t &gt; x$\uff0c\u5373 $x &lt; t &lt; 0$\uff0c\u5bf9\u5e94 $0 &lt; d &lt; -x$\u3002<\/li>\n\n\n\n<li>\u82e5 $t = 0$\uff1a$f(0) = 0 &lt; 2^x$\uff0c\u4e0d\u6ee1\u8db3\u3002<\/li>\n\n\n\n<li>\u82e5 $t &gt; 0$\uff1a$f(t) = -2^{-t} &lt; 0 &lt; 2^x$\uff0c\u4e0d\u6ee1\u8db3\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\boxed{x &lt; 0 \\implies D(x) = (0,\\;-x)}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u60c5\u5f62\u4e8c\uff1a$x &gt; 0$\u3002<\/strong> \u6b64\u65f6 $f(x) = -2^{-x} \\in (-1, 0)$\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u82e5 $t &lt; 0$\uff1a$f(t) = 2^t &gt; 0 &gt; -2^{-x}$\uff0c<strong>\u4e00\u5b9a\u6ee1\u8db3<\/strong>\u3002\u6761\u4ef6 $t &lt; 0$ \u5373 $d &lt; -x$\u3002<\/li>\n\n\n\n<li>\u82e5 $t = 0$\uff1a$f(0) = 0 &gt; -2^{-x}$\uff0c\u6ee1\u8db3\u3002\u6b64\u65f6 $d = -x$\u3002<\/li>\n\n\n\n<li>\u82e5 $t &gt; 0$\uff1a$-2^{-t} &gt; -2^{-x} \\iff 2^{-t} &lt; 2^{-x} \\iff t &gt; x$\uff0c\u5373 $d &gt; 0$\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\boxed{x &gt; 0 \\implies D(x) = (-\\infty,\\;-x\\,] \\cup (0,\\;+\\infty)}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.3 \u5206\u60c5\u51b5\u8bc1\u660e D(x<sub>2<\/sub>) \u2286 D(x<sub>1<\/sub>)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u4e8e $x_1 x_2 \\neq 0$\uff0c\u6709\u56db\u79cd\u60c5\u51b5\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>$x_1 &lt; 0$ \u4e14 $x_2 &lt; 0$\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$f(x_1) \\leq f(x_2) \\iff 2^{x_1} \\leq 2^{x_2} \\iff x_1 \\leq x_2$\uff0c\u6240\u4ee5 $-x_2 \\leq -x_1$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$D(x_2) = (0, -x_2) \\subseteq (0, -x_1) = D(x_1). \\quad$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>$x_1 &gt; 0$ \u4e14 $x_2 &gt; 0$\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$f(x_1) \\leq f(x_2) \\iff -2^{-x_1} \\leq -2^{-x_2} \\iff x_1 \\leq x_2$\uff0c\u6240\u4ee5 $-x_2 \\leq -x_1$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$(-\\infty, -x_2] \\subseteq (-\\infty, -x_1]$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e24\u8005\u90fd\u542b $(0, +\\infty)$\uff0c\u6240\u4ee5 $D(x_2) \\subseteq D(x_1)$\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>$x_1 &lt; 0$ \u4e14 $x_2 &gt; 0$\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b64\u65f6 $f(x_1) = 2^{x_1} &gt; 0$ \u800c $f(x_2) = -2^{-x_2} &lt; 0$\uff0c\u6240\u4ee5 $f(x_1) &gt; f(x_2)$\uff0c<strong>\u6b64\u60c5\u5f62\u4e0d\u51fa\u73b0\u3002<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>$x_1 &gt; 0$ \u4e14 $x_2 &lt; 0$\u3002<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$f(x_1) &lt; 0 &lt; f(x_2)$\uff0c\u6240\u4ee5 $f(x_1) \\leq f(x_2)$ \u6210\u7acb\u3002\u9700\u8bc1 $D(x_2) \\subseteq D(x_1)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$D(x_2) = (0, -x_2), \\quad D(x_1) = (-\\infty, -x_1] \\cup (0, +\\infty)$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4efb\u53d6 $d \\in D(x_2) = (0, -x_2)$\uff0c\u663e\u7136 $d &gt; 0$\uff0c\u6240\u4ee5 $d \\in (0, +\\infty) \\subseteq D(x_1)$\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bc1\u6bd5\u3002$\\blacksquare$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">\u7b2c(3)\u95ee\uff1a\u554a\u5440\uff0c\u9a87\u6b7b\u6211\u529b<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u73b0\u5728 $f(x)$ \u4e0d\u518d\u5177\u4f53\u7ed9\u51fa $x \\geq 0$ \u7684\u8868\u8fbe\u5f0f\uff0c\u53ea\u4fdd\u7559\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u5df2\u77e5<\/strong>\uff1a$x &lt; 0$ \u65f6 $f(x) = 2^x$\u3002<\/li>\n\n\n\n<li><strong>\u6761\u4ef6\u2460<\/strong>\uff1a\u82e5 $f(x_1) \\leq f(x_2)$\uff0c\u5219 $D(x_2) \\subseteq D(x_1)$\u3002<\/li>\n\n\n\n<li><strong>\u6761\u4ef6\u2461<\/strong>\uff1a\u5f53 $0 &lt; x &lt; 1$ \u65f6\uff0c$f(x) &lt; f(0)$\u3002<\/li>\n<\/ul>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\">\u6ce8\u610f\u6761\u4ef6\u2460\u6bd4\u7b2c(2)\u9898\u7684\u7ed3\u8bba\u66f4\u5f3a\uff1a\u5b83\u53bb\u6389\u4e86 $x_1 x_2 \\neq 0$ \u7684\u9650\u5236\uff0c\u5bf9<strong>\u6240\u6709<\/strong>\u5b9e\u6570\u90fd\u6210\u7acb\u3002<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\">3(i)\uff1a\u8bc1\u660e f(0) &gt; 1<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u601d\u8def<\/strong>\uff1a\u53cd\u8bc1\u6cd5\u3002\u5982\u679c $f(0) &lt; 1$\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u8d1f\u534a\u8f74\u4e0a $2^x \\to 1^-$ \u627e\u5230\u77db\u76fe\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u53cd\u8bbe<\/strong> $f(0) &lt; 1$\u3002\u56e0\u4e3a $\\lim_{x \\to 0^-} 2^x = 1$\uff0c\u5b58\u5728 $a &lt; 0$\uff0c\u4f7f\u5f97<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(0) &lt; 2^a = f(a)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5373 $f(0) \\leq f(a)$\u3002\u7531\u6761\u4ef6\u2460\uff0c\u4ee4 $x_1 = 0$\uff0c$x_2 = a$\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$D(a) \\subseteq D(0) \\tag{$\\star$}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u73b0\u5728\u6784\u9020\u4e00\u4e2a\u5c5e\u4e8e $D(a)$ \u7684\u5143\u7d20\u3002\u53d6 $d$ \u6ee1\u8db3 $0 &lt; d &lt; \\min\\{-a,\\;1\\}$\uff0c\u5219\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$a + d &lt; a + (-a) = 0 \\implies a+d &lt; 0$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $f(a+d) = 2^{a+d} &gt; 2^a = f(a)$\uff0c\u5373 $d \\in D(a)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531 $(\\star)$\uff0c$d \\in D(0)$\uff0c\u5373 $f(0 + d) &gt; f(0)$\uff0c\u4e5f\u5c31\u662f $f(d) &gt; f(0)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f46\u662f $0 &lt; d &lt; 1$\uff0c\u7531\u6761\u4ef6\u2461 $f(d) &lt; f(0)$\uff0c<strong>\u77db\u76fe<\/strong>\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5\u5047\u8bbe\u4e0d\u6210\u7acb\uff0c$\\boxed{f(0) \\geq 1}$\u3002$\\blacksquare$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">3(ii)\uff1a\u8bc1\u660e f(x) \u5728 (0, +\u221e) \u4e0a\u5355\u8c03\u9012\u589e<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u5413\u54ed\u4e86\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">\u7b2c\u4e00\u6b65\uff1a\u5f53 $0 &lt; x &lt; 1$ \u65f6\uff0c$f(x) \\leq 0$<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u53cd\u8bbe<\/strong>\u5b58\u5728 $x_0 \\in (0,1)$ \u4f7f\u5f97 $f(x_0) &gt; 0$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u6761\u4ef6\u2461\uff0c$f(x_0) &lt; f(0)$\uff0c\u6240\u4ee5 $-x_0 \\in D(x_0)$\uff08\u56e0\u4e3a $f(x_0 + (-x_0)) = f(0) &gt; f(x_0)$\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u4e8e $\\lim_{t \\to -\\infty} 2^t = 0$\uff0c\u53ef\u53d6 $a &lt; 0$ \u4f7f\u5f97 $2^a \\leq f(x_0)$\uff0c\u5373 $f(a) \\leq f(x_0)$\u3002\u7531\u6761\u4ef6\u2460\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$D(x_0) \\subseteq D(a)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $-x_0 \\in D(a)$\uff0c\u5373 $f(a &#8211; x_0) &gt; f(a)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f46 $a &#8211; x_0 &lt; a &lt; 0$\uff0c\u6240\u4ee5 $f(a-x_0) = 2^{a-x_0} &lt; 2^a = f(a)$\uff0c<strong>\u77db\u76fe\uff01<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $0 &lt; x &lt; 1 \\implies f(x) \\leq 0$\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">\u7b2c\u4e8c\u6b65\uff1a\u5bf9\u6240\u6709 $x &gt; 0$\uff0c$f(x) \\leq 0$<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u53cd\u8bbe<\/strong>\u5b58\u5728 $x_0 > 0$ \u4f7f\u5f97 $f(x_0) > 0$\u3002\u53d6 $s \\in (0, \\min\\{x_0, 1\\})$\uff0c\u4ee4 $h = x_0 &#8211; s > 0$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53d6 $y &lt; 0$ \u4f7f\u5f97 $2^y &lt; f(x_0)$\uff08\u5373 $f(y) &lt; f(x_0)$\uff09\u3002\u4ee4 $d = x_0 &#8211; y &gt; 0$\uff0c\u5219<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(y + d) = f(x_0) &gt; f(y) \\implies d \\in D(y)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a5\u4e0b\u6765\u8003\u5bdf $y &#8211; h$\u3002\u56e0\u4e3a $y &lt; 0$ \u4e14 $h &gt; 0$\uff0c\u6240\u4ee5 $y &#8211; h &lt; y &lt; 0$\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(y &#8211; h) = 2^{y-h} &lt; 2^y = f(y)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u6761\u4ef6\u2460\uff08$f(y-h) \\leq f(y)$\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$D(y) \\subseteq D(y-h)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $d \\in D(y-h)$\uff0c\u5373<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(y &#8211; h + d) &gt; f(y-h)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8ba1\u7b97\u6307\u6807\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$y &#8211; h + d = y &#8211; h + (x_0 &#8211; y) = x_0 &#8211; h = s$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $f(s) &gt; f(y-h) = 2^{y-h} &gt; 0$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f46 $0 &lt; s &lt; 1$\uff0c\u7531\u7b2c\u4e00\u6b65 $f(s) \\leq 0$\uff0c<strong>\u77db\u76fe\uff01<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6240\u4ee5 $\\boxed{\\forall\\, x &gt; 0,\\; f(x) \\leq 0}$\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">\u7b2c\u4e09\u6b65\uff1a$f$ \u5728 $(0, +\\infty)$ \u4e0a\u4e25\u683c\u9012\u589e<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u4efb\u53d6 $0 &lt; a &lt; b$\uff0c\u4ee4 $h = b &#8211; a &gt; 0$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u7b2c\u4e8c\u6b65\uff0c$f(a) \\leq 0$\u3002\u800c $f(-h) = 2^{-h} &gt; 0$\u3002\u6240\u4ee5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(a) \\leq 0 &lt; 2^{-h} = f(-h)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7531\u6761\u4ef6\u2460 $D(-h) \\subseteq D(a)$\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53c8\u7531 3(i)\uff0c$f(0) \\geq 1 &gt; 2^{-h} = f(-h)$\uff0c\u6240\u4ee5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(-h + h) = f(0) &gt; f(-h) \\implies h \\in D(-h)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u56e0\u6b64 $h \\in D(a)$\uff0c\u5373<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$f(a + h) &gt; f(a) \\implies f(b) &gt; f(a)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5bf9\u4efb\u610f $0 &lt; a &lt; b$ \u90fd\u6709 $f(a) &lt; f(b)$\uff0c\u5373$f(x) \\text{ \u5728 } (0, +\\infty) \\text{ \u4e0a\u4e25\u683c\u5355\u8c03\u9012\u589e\u3002}$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u663e\u7136\u53ef\u5f97\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\boxed{f(x) \\text{ \u5728 } (0, +\\infty) \\text{ \u4e0a\u5355\u8c03\u9012\u589e\u3002}} \\quad\\blacksquare$$<br><br>\u6ca1\u6709\u4eba\u8bdd\u7248\u672c\u3002\u770b\u4e0d\u61c2\u4e0a\u8ff0\u8fc7\u7a0b\u7684\u8bf7\u4e0d\u8981<strong>\u5e7b\u60f3<\/strong>\u505a\u51fa\u8fd9\u9053\u9898\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u7ed3\u8bed<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u5199\u4e0d\u51fa\u6765\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u603b\u4e4b\uff0c\u4f60\u4eec21\u5c4a\u8001\u5b66\u957f\u6ca1\u5403\u8fc7\u8fd9\u6837\u7684\u82e6\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u795d\u597d\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>19. (17\u5206) \u5df2\u77e5\u51fd\u6570 $f(x)$ \u7684\u5b9a\u4e49\u57df\u4e3a $\\mathbf{R}$\uff0c\u4e14\u5f53 $x&lt;0$ \u65f6\uff0c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[21,23],"class_list":["post-499","post","type-post","status-publish","format-standard","hentry","category-6","tag-21","tag-23"],"_links":{"self":[{"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/posts\/499","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=499"}],"version-history":[{"count":34,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/posts\/499\/revisions"}],"predecessor-version":[{"id":534,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=\/wp\/v2\/posts\/499\/revisions\/534"}],"wp:attachment":[{"href":"https:\/\/blog.hcha.site\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=499"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=499"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.hcha.site\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}