https://cn-sunisalex-pages.zh-cn.edgeone.cool/categories/%E9%AB%98%E8%81%94%E4%B8%80%E8%AF%95/Recent content in 高联一试 on ∇Alex的实验室Hugozh-CNMon, 09 Mar 2026 20:51:00 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/concavity-and-convexity-of-functions/Mon, 09 Mar 2026 20:51:00 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/concavity-and-convexity-of-functions/<h1 id="函数的凹凸性-convexity-and-concavity">函数的凹凸性 (Convexity and Concavity)</h1> <h1 id="前言同一事物三种视角">前言：同一事物，三种视角</h1> <p>什么是函数的<strong>凹凸性</strong>？当我们面对这个问题时，得到的答案往往取决于提问的语境。这有点像费曼在《费曼物理学讲义》中所说的：</p> <blockquote> <p>科学是对同一事物不同角度的认识。</p>https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/constrained-extremum-problem001/Fri, 06 Mar 2026 21:56:50 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/constrained-extremum-problem001/<h2 id="题目">题目</h2> <p>已知 <span class="math math-inline"> $x, y \ge 0$ </span>，且满足方程 <span class="math math-inline"> $x + 4y = x^2 y^3$ </span>，求 <span class="math math-inline"> $(\frac{8}{x} + \frac{1}{y})_{\min}$ </span> 的值。</p> <hr> <h2 id="解答过程人类阵营">解答过程(人类阵营)</h2> <h3 id="消元单变量基本不等式">消元+单变量基本不等式</h3> <ol> <li> <p><strong>将原方程变形为关于 <span class="math math-inline"> $x$ </span> 的一元二次方程：</strong> </p> <div class="math math-block"> $$y^3 x^2 - x - 4y = 0$$ </div></li> <li> <p><strong>利用求根公式解出 <span class="math math-inline"> $x$ </span>（取正根）：</strong> </p>