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cyrxdzj committed Oct 18, 2023
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2 changes: 1 addition & 1 deletion 2021-07-02-LuoguMonthlyCompetition202105Summary/index.html
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2021-07-07-HowToUseLuoguCard/index.html
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Expand Up @@ -923,7 +923,7 @@ <h2 id="镜像仓库"><a href="#镜像仓库" class="headerlink" title="镜像
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2021-07-09-Grade7Term2FinalExamSummary/index.html
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Expand Up @@ -882,7 +882,7 @@ <h3 id="十、总分与排名"><a href="#十、总分与排名" class="headerlin
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2021-07-09-Grade7Term2MidtermExamSummary/index.html
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2021-07-22-LuoguMonthlyCompetition202107Summary/index.html
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Expand Up @@ -1189,7 +1189,7 @@ <h3 id="五、总分"><a href="#五、总分" class="headerlink" title="五、
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2023-05-21-TalkingAboutComputerSecurityFromTheIntelligentClassromSystem/index.html
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Expand Up @@ -858,7 +858,7 @@ <h3 id="解决问题的建议"><a href="#解决问题的建议" class="headerlin
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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13 changes: 10 additions & 3 deletions 2023-09-22-CyrxNote-Math-001/index.html
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Expand Up @@ -454,7 +454,7 @@ <h1 class="description center-align post-title">cyrxdzj的文化课学习笔记

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<i class="far fa-file-word fa-fw"></i>文章字数:&nbsp;&nbsp;
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Expand Down Expand Up @@ -512,7 +512,14 @@ <h3 id="4、使系数相等"><a href="#4、使系数相等" class="headerlink" t
<p>那么，我们考虑改变一下 $ab$ 这边。给 $a$ 搞个系数 $2$，给 $b$ 搞个系数 $3$，最后统一 $\div6$，我们就有 $ab=(2a\cdot3b)\times\frac{1}{6}\le(\frac{2a+3b}{2})^2\times\frac{1}{6}=6$，那么 $\frac{1}{ab}$ 的最小值就是 $\frac{1}{6}$。</p>
<h3 id="5、将分数中的分子用类似于带分数的思想提取出来"><a href="#5、将分数中的分子用类似于带分数的思想提取出来" class="headerlink" title="5、将分数中的分子用类似于带分数的思想提取出来"></a>5、将分数中的分子用类似于带分数的思想提取出来</h3><p>当 $x&gt;1$ 时，求 $\frac{x^2-x+1}{x-1}$ 的最小值。</p>
<p>对于这道题，可以发现 $(x-1)x=x^2-x$，于是原式可以被化为 $\frac{x(x-1)+1}{x-1}=x+\frac{1}{x-1}$。接着再用配凑法，即可算出式子的最小值为 $3$。</p>
<p>是的，多项式也可以除以多项式的，不过可能会有余数。$(x^2-x+1)\div(x-1)=x……1$。</p>
<p>提取什么东西应该根据题目要求来。例如，同样是 $x&gt;1$ 时，求 $\frac{x^2-2x+5}{x-1}$ 的最小值。</p>
<p>可以注意到，此时最好提取 $x-1$ 出来，得 $x-1+\frac{4}{x-1}$。可以很轻松地算出最小值为 $4$，当且仅当 $x-1=\frac{4}{x-1}$ 时，即 $x=3$ 时，等号成立。</p>
<p>扩展一个知识：整式除法。</p>
<p>例如，当 $x&gt;-2$，求 $\frac{2x^2+8x+20}{x+2}$。</p>
<p>首先，当前被除数是 $2x^2+8x+20$，当前除数是 $x+2$。</p>
<p>可以发现，当前被除数的最高次数项是 $2x^2$，系数是 $2$，次数是 $2$。而除数 $x+2$ 的最高次数项的系数是 $1$，次数是 $1$。</p>
<p>因此可以发现，如果将 $x+2$ 乘上 $2x$ 可以得到 $2x^2+4x$。因此，$\frac{2x^2+8x+20}{x+2}=2x+\frac{(2x^2+8x+20)-(2x^2+4x)}{x+2}=2x+\frac{4x+20}{x+2}$。被除数中的 $2x^2$ 就被消去了，当前被除数就变成了 $4x+20$。</p>
<p>接着，当前被除数的最高次数项是 $4x$，系数是 $4$，次数是 $1$。当前除数的最高次数项的系数和次数都没变，都还是 $1$。因此，$4(x+2)=4x+8$，$2x+\frac{4x+20}{x+2}=2x+4+\frac{(4x+20)-(4x+8)}{x+2}=2x+4+\frac{12}{x+2}$。化简成这样，再用基本不等式就不难了。</p>
<h3 id="6、转化为一元二次不等式求解"><a href="#6、转化为一元二次不等式求解" class="headerlink" title="6、转化为一元二次不等式求解"></a>6、转化为一元二次不等式求解</h3><p><a href="/2023-10-04-CyrxNote-Math-003">一元二次不等式的解法在此。</a></p>
<p>已知 $ab+a+b=8,a&gt;0,b&gt;0$，则：</p>
<p>A. $ab$ 最大值为 $2$</p>
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2023-09-22-CyrxNoteIndex/index.html
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Expand Up @@ -888,7 +888,7 @@ <h1 class="description center-align post-title">cyrxdzj的文化课学习笔记
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
class="white-color">16.9k</span>&nbsp;字
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2 changes: 1 addition & 1 deletion 2023-09-23-CyrxNote-Math-002/index.html
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Expand Up @@ -920,7 +920,7 @@ <h5 id="5-3、集合有交集但不要求互相包含"><a href="#5-3、集合有
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2023-09-28-CyrxNote-Chemistry-001/index.html
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Expand Up @@ -1009,7 +1009,7 @@ <h3 id="花絮"><a href="#花絮" class="headerlink" title="花絮"></a>花絮</
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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15 changes: 12 additions & 3 deletions 2023-10-04-CyrxNote-Math-003/index.html
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Expand Up @@ -454,7 +454,7 @@ <h1 class="description center-align post-title">cyrxdzj的文化课学习笔记

<div class="info-break-policy">
<i class="far fa-file-word fa-fw"></i>文章字数:&nbsp;&nbsp;
1.2k
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Expand Down Expand Up @@ -512,6 +512,15 @@ <h3 id="2、分类讨论"><a href="#2、分类讨论" class="headerlink" title="
<p><img src="../post_image/CyrxNote-Math-003/CyrxNote-Math-003-07.PNG" alt=""></p>
<p>当 $m&gt;2$ 时，$\frac{2}{m}&lt;1$。此时解集为 $\frac{2}{m}\le x\le1$。</p>
<p><img src="../post_image/CyrxNote-Math-003/CyrxNote-Math-003-08.PNG" alt=""></p>
<p>可以发现，一般这种类型的一元二次不等式，会分这么几类讨论：</p>
<ol>
<li><p>当一元二次不等式的二次项系数为 $0$ 导致原式退化为一元一次不等式时。此时就需要用解一元一次不等式的方法处理。</p>
</li>
<li><p>当一元二次不等式的零点只有一个时。此时就判断该不等式是否允许取等号。如允许，则解集为一个数（例如上面的 $x=1$）或解集为全集（不等式恒成立）；如不允许，则可能无解，或解集为除了这个数的所有数。</p>
</li>
<li><p>当一元二次不等式的零点有两个时。此时就要考虑图象的开口方向和零点的大小关系，根据具体情况写出解集。</p>
</li>
</ol>
<p>一个看似简单的不等式，分了 5 类作答……在数学中养成细心审题、全面考虑的习惯还是很重要的。</p>
<h3 id="3、一元一次不等式与一元二次不等式的分类讨论结合"><a href="#3、一元一次不等式与一元二次不等式的分类讨论结合" class="headerlink" title="3、一元一次不等式与一元二次不等式的分类讨论结合"></a>3、一元一次不等式与一元二次不等式的分类讨论结合</h3><p>若关于 $x$ 的不等式 $ax-b\le0$ 的解集为 {$x|x\ge2$}，则关于 $x$ 的不等式 $ax^2+(3a-b)x-3b&lt;0$ 的解集是？</p>
<p>根据这个一元一次不等式的解集，我们可以发现，$ax-b$ 的值随 $x$ 的升高而<strong>降低</strong>。换句话说，$a&lt;0$。</p>
Expand Down Expand Up @@ -596,7 +605,7 @@ <h3 id="7、一元二次不等式恒成立的问题"><a href="#7、一元二次
<p>综上，$-2\sqrt2&lt;a&lt;2\sqrt2$。</p>
<p>再看看这个：$\frac{2x^2+2mx+m}{4x^2+6x+3}&lt;1$ 对所有实数 $x$ 恒成立，则实数 $m$ 的取值范围是？</p>
<p>首先，因为 $6^2-4\times4\times3=-12&lt;0$，可以知道 $4x^2+6x+3$ 始终大于 $0$。因此，可以放心地将分母移项到右边，再稍微整理一下，得 $0&lt;2x^2+(6-2m)x+(3-m)$。</p>
<p>结合上表，可知 $\Delta=(6-2m)^2-4\times2\times(3-m)=4m^2-24m+36-24+8m=4m^2-16m+12=4(m-1)(m-3)&lt;0$，解得 $1&lt;m&lt;3$。</p>
<p>结合上表，可知 $\Delta=(6-2m)^2-4\times2\times(3-m)\\=4m^2-24m+36-24+8m\\=4m^2-16m+12\\=4(m-1)(m-3)&lt;0$，解得 $1&lt;m&lt;3$。</p>
<h3 id="8、在限定-x-的取值范围的前提下，一元二次函数的最值"><a href="#8、在限定-x-的取值范围的前提下，一元二次函数的最值" class="headerlink" title="8、在限定 $x$ 的取值范围的前提下，一元二次函数的最值"></a>8、在限定 $x$ 的取值范围的前提下，一元二次函数的最值</h3><p>我们都知道，当 $x$ 的取值范围不限定时，一元二次函数要么只有最大值，要么只有最小值。但是限定了 $x$ 的取值范围后，一元二次函数就有可能同时拥有最小值和最大值了。</p>
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2023-10-09-CyrxNote-Math-004/index.html
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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2 changes: 1 addition & 1 deletion 2023-10-12-CyrxNote-Physics-001/index.html
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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&nbsp;<i class="fas fa-chart-area"></i>&nbsp;站点总字数:&nbsp;<span
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